Anyone know what Army Regulation (or any branch) that allows a higher-ranking officer to call a lower-ranking officers by their first name?

I've never subscribed to the practice of using someone's first name in uniform. I give the Junior ranks the respect of their rank...just like me, they earned it. I've had officers refer to me by my first name, it makes my uncomfortable, and some of them don't deserve the privilege.
I find it is just more appropriate when in uniform to be IN uniform.

I do, however, refer to other NCOs of my rank by their first names, but only if I know them pretty well. But just try and call an officer by his first name...I never have, but I have been tempted when they call me by mine.

Nope it is a mark of the highter to the lower.
"Jimmy" the employer said to his driver, Be here at 4;00 . Yes Mr. Jones replied Jimmy

As an enlisted Marine (Gunny) who had been referred to by his first name by senior officers, I viewed that as a sign of trust and respect and was honored. Of course, what did I call them in return? ...... Sir. Semper Fi.

After my three year stint in the army I joined Texas Department of Criminal Justice. During my employment there everyone was addressed by their surname and if you held a supervisory position, then your title was included. After four and half years I left to join the Federal Bureau of Prisons. Part of the culture shock was that staff regardless of position addressed each other by their first names. What I found was that how you carry yourself determines how your peers and those who you supervise respond to your directives.

Seem to me that if you have to ask the question the answer should be obvious; no regulation needed. Aside from Article 134, or 92 of the UCMJ each service has their own customs and traditions but personally, if I were a CW4 and I had any desire to see CW5 I'd be calling a senior officer by their rank or Sir. As Capt. Seid Waddell said, "I was on a first-name basis with my commander on my first assignment. He called me Seid and I called him Sir.") I think that about covers it. It's been fifty some years since I was in but "familiarity breeds contempt" was the saying then and I suspect it still is; if you need proof just look at the current CinC.

My memory of the "Officer's Guide" was that the superior never remembers the difference in rank and the subordinate never forgets it. Served me well for 13 years in the military. One time I pulled out a cigarette (I smoked back then) and a flame appeared at the end of it attached to a 1 star general. Spoke volumes.

This is speaking strictly about my situation. I served under a Brigadier General that served with my father when they were in the army guard in 1936. He was a colonel then. When I served for a brief time in the guard it just happened to be under this family friend. He knew who I was. He was inspecting my unit one day and as usual was surrounded by all types of officers. I just happened to be walking by and was with several others we rendered the proper salute as we were passing. Then this loud voice called out SGT MARTIN!! I stopped and did about face and saluted again. The General had decided to have a friendly discussion with me. The look on the officers and enlisted faces was priceless. No one could figure out why the General was talking to a lowly Sargent and not to them. After about five minutes of talking I snapped to saluted and left to go back to my tank. Years later after he retired and I was out of the service I interned in his law office. Every one in the office used his christian except me. During that conversation back in the guard he told me I could call him by his first name...I asked what it was and he said General. The entire time I worked for him and was around him that's what I called him.

I was in usaf until 2001. This was never an issue. Everyone was rank and last name period. No question about when it is appropriate since this is alway right. Best not to get into the habit of first name then have to think when it is appropriate.

During my twenty-two years of service, one of my assignments was the 7th Inf Div, Co C Ft Ord, CA, as the Company Clerk I interacted with my 1SG Victor C. Underwood (quite a character and unquestionably a Warrior) numerous times throughout the day. Our offices were connected but walled off with a small "door-less" doorway for access between our offices. Whenever TOP would call me he would refer to me as SPC McCausland because there was another soldier in his office or he didn't know if there was someone in my office but when we were alone or on our Friday Night NCO Club ( yes, I know SPC in an NCO club, that's another story altogether) outings it was always Mac. We were both Vietnam Veterans and after a few double shots of JD, TOP would open up with the most amazing war stories.
Our experiences as far as facing the enemy in combat were as different as night and day.
If you look up Victor C. Underwood and read his DSC Citation you will see a "real" warrior.
So, yes officially and keeping with military protocol my ISG never broke rank and called me Specialist but when we were alone, we were two Nam soldiers and buddies and it was always "Mac".

Most of my superiors called me by first name I never called them by first name out of respect