How would you react to an E2 who "smart mouths" you in formation?

The private E-2, soon to be E-1, should receive article 15, and every shit detail possibly for 30 days.

I would pull him out of formation.
Take him or her to the back of the formation.
Put them in the front leaning rest, and tell them not to move. Have his immediate supervisor stand next to him,Until you can come back after the formation.
Give them there work assignment again.
Ask them if they understand the instructions, Do they have any questions.
Then ask them if they know why they are in the push-up position.
If they do not. Enlighten them.
Inform them that they will be formerly counselled, More than likely be given a ARTICLE 15.
And after enough of them the separation paperwork will be started.
To put them out of the Military and they will not be eligible for any V.A. Benefits. Such as V.A. Medical College programs, And government jobs. If this person received a enlistment bonus That will be taken back.
That should straighten this person out.

Okay everyone. we're all brothers in arms. Let's agree to disagree on this one. There are a lot of parameters not revealed here and let's be honest. As most of us would love to greet this recruits bad attitude with a swift and powerful boot to his rear. Times methods and meas of correction in the military have changed dramatically over the years. It is quite obvious this recruit saw something witnessed something in his leadership that led him/her to believe it was okay to do what he/her did. I have no doubt if we posed this question to 100 members from every branch, we'd get 100 different options from every branch. We are all trained experts in our respective fields and have a wealth of knowledge to lend especially in situations such as this. Truth is, we'd all take tremendous pleasure in strategically advancing boot to head on this disrespectful young lad. Leadership would have to be addressed first on this one! I've spent more than twenty (20) years in service and have served alongside Navy Marines Air Force security forces and Army Air Borne. I've never seen a good leader blatantly disrespected. It starts at the top, but the end results come from the bottom. As a ground troops leader, Ultimate responsibility begins and Ends with YOU.

Put them in the front leaning rest position.. Make them remember the chain of command

It depends on the era and the command climate. In the Days of Olde, there would have been immediate, physical corrective action of a personal nature directly between E-5 and E-2. Those days are long gone. One must be first, certain that the NCO will be backed by his superior Chan of command; and second within guidelines of policy. It is increasingly difficult to administer correction to an individual without escalating to some administrative action which does not allow for a change of heart in the offender (e.g. the replacement of a PT thrashing by an NCO for an Accidental Discharge with an administrative punnishment with brigade JAG approval for a 'Negligent Discharge in the Army).
Perhaps an approach of making the offender's entire squad or Fire Team perform some sort of additional training would bring peer pressure to bear, but beware the command climate. Even that can backfire.

You are the person in charge Sgt, a professional leader in this case. So you tell this unrespectful obvious dumbass "bad apple" to hold still while you finish with handing out the tasks to your troops! Your primary task is complete. Now you can deal with this disciplinary problem. Stay cool and bring the E2 to your office out of sight of the other troops. Now you can deal with his real issues toward respect for authority and taking orders. If there is a supervisor of this person, you include them in the reprimand actions. In no way would I let the subordinate dictate the situation or have chance to act in control. Have the person stand at attention the whole time and then read them the UCMJ that tells them that they have just committed a serious offense and can be punished by hard labor demoted or both even kicked out of the military! He may want you to kick him/her out, but that is not always best for military. So it is your job to restore order and regain his respect. How do you think you can do that is what leadership is all about! Use your head, not just anger like most bosses in military do! It is hard to do but it can turn a bad apple into to outstanding leaders. There are many examples in our corps that prove it! As a TSgt in AF, I was given a whole flight of unwanted misfits (young men & women ) like you described that nobody else wanted to deal with. I spoke to them each individually and asked them to explain their situation and how they got sent to my flight. I learned that many of them had authority issues and nobody really listened to them so they ranted to get attention and be noticed. Their supervisors took it as lazinessness, disrespect and unable to adapt t military standards. I saved six from getting disciplined and the others I helped out of the service for other reasons! Being professional and in control at all times as a person in charge, is very important with the types of troops we get from society today. They don't have one shred of self-discipline or self-control so we must teach it to them through making them part of the team, sometimes that takes having them do extra labor and exercises for acts of disobedience. My friend, an Army Ranger then AF MSgt had his mechanic troops & himself do army runs with packs and meet on weekends to clean-up the overdue mx on out of service generators by end of 6 months, he had the best maintenance flight in our unit. That was truly great leadership and team-building! They were the best at everything assigned them under his command! I tried to learn from his example and experience and now I passed on to you for good application and turning troops around!

Praise in public discipline in private. Have Soldier stand by after formation get to bottom of issue. Then get it in writing and ensure the punsihment goes with the crime

Intro him/her to wall-to-wall counseling

I wouldn’t be posting it to social media looking for others “opinions”. As an NCO you get paid to LEAD. There will always be a smart mouthed kid trying to challenge your authority, regardless of your rank. Find a Senior NCO to model your reaction to these situations.

Wouldn't be the first time I've heard or experienced that. My reaction is usually the same:
Pvt, are you really going to sit here, in front of your team/buddies/friends/colleagues/etc, and tell them and me that you think you're better than them and not only don't have to work, but that I should give them your share of the load? Are you, a pvt, also disobeying a direct order by superior? Are you, still a pvt, telling me that you think you've done enough in your 10 minute career to warrant being excused from the work that every other private in the history of (insert branch here) has had to do, myself included? Are you, once again still a pvt, telling me that you need to be chaptered out of the (insert branch here) for dereliction of duty and failure to adapt? Would you like to discuss your views on this matter with 1SG (or branch equivalent)?
When I start asking questions to blow holes in their extremely flawed logic and sense of entitlement, that usually creates a change of heart.
I had one incident where I took the rank off my chest and handed it to the individual. I informed him that since he believed us to be equal (in rank, not as human beings) that he should take it. He was visibly uncomfortable and started backpedaling very quickly.