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

Insubordination toward an office or NCO is an UCMJ article 15 offence. I feel there are a number of ways that this could be handles; straight away go the article 15 route, if this is a habitual response to an NCO or officer. Depending on the situation i.e. at first time offence, my preference would be first to handle a verbal reprimand in private or a written reprimand depending on the severity and lastly an article 15. I maybe over simplifying the process, I am just trying to make the point that insubordination cannot be tolerated.

Things sure have changed since my military days. During early Vietnam era 62 to 66 The shithead at least would be assigned every crap detail I could find. And in between I would see he pulled Guard detail and got PTd damn near to death. If that didn’t straighten him out there was always arrival 15 or the stockade.

First I retired in 2013. When I was put in charge of a new section, I would wrestle the biggest soldier (all in fun). I had no issues losing skin on black top, my goal was winning. That set a standard for my toughness. Then I took a list of their issues and fixed what I could. I was hard on them and they knew it, but I was clear with what I expected of them. I set the example for them to follow, and took care of them when I could.

To the Pvt mouthing, I would have chewed and smoked the hole section. My section would correct him on their own. If that does not work then you counsel amd inform the individual of their failures.
Then last I checked, UCMJ states that you may physically aid a soldier to follow your orders. That does not mean beat them, be creative and teach a lesson.

Sadly in reallity, you have to counsel, counsel, counsel and pray officers follow through.

Insubordinate, should be dressed down by 1st. Sargent or officer in charge. Given extra duty and put on notice of future disablement if attitude does not improve.

Lesson #1
You are in the motor pool on Friday, 1630 hours for closeout formation. Your platoon sergeant tells you to sweep the line prior to release. You salute him sharply, about face to your squad, and tell Jones to grab a broom and sweep the line. Jones says "nope, I wont do it" Jones, I'm giving you a lawful directive to sweep the line. Jones say no and you can't make me. So what, squad leader, do you do?
Two basic responsibilities for the NCO is the welfare of my soldiers and accomplishing the mission, remember....
Okay you say, Miller, grab the broom and sweep the line. Squad, you assist and when complete I'll see you for PT on Monday. Jones, my office now. We must discuss your behavior. of course you know the DA4856 and so on.....maybe a recommendation for a BAR or whatever. So....the line was swept. Mission accomplished. And you attempt to preserve Jones's record with discipline.
I hope my spelling is proper. I learned this a long time ago from an old CSM with like 33 years in our Army. So don't forget it troop. You can do both if you think before planting foot in ass! God bless the troopers.

1. As the Senior Leader in the situation, don't be drawn into the punk's BS. Calm will win.
2. At first actual refusal by Pvt Snuffy, ask solider if he/she is disobeying a lawful order from an NCO.
3. My guess is Pvt will comply to directive. Prepare written counseling for Private, present prior to COB, same day, with Squad Leader present. Inform 1SG/Pltt. Sgt. of resolution. Regardless of compliance with original directive
If Pvt really derails, at step 3. it's into the office we go. Immediate written counseling statement with UCMJ action pending bullet point. Do this part correctly Leaders! JAG throws out lots of Art 15's because first line leaders don't get it right. This is a teachable moment for the next NCOPD/OPD.
This situation has/will happen to every leader in every branch at some point. Happened to me.

Excercise them till they comply !

he would be subjected to the decision of the commander and made an example to his peers

An reprimand in writing should change his behavior. If not an Article 15 in his military records will certainly get his attention.

I would call the entire group to attention and tell them this: "... any organization is only as good as its weakest link. No one gets to choose what they will or will not do in this unit. The individual in question, is placed on restriction and will be referred for violations of the UCMJ for refusing a direct order. However, in the interim, someone else in this unit will now have to do his job in addition to their own. Thus lowering the Combat efficiency of the unit. You can all think about that and ask yourself one question: "... Why should he ignore orders and think of himself first - why we have to take up the slack..." Because, in Combat, -the soldier that refuses to do his job gets others killed.
Just remember that "... Duty is the absence of self..."