Damn, that scene is fantastic.
But the scene that has stayed with me for 12 years is a scene from ER in which no one even says a thing. It's from the very end of the season 6 finale titled "May Day." John Carter's coworkers have confronted him about his escalating drug use and, after breaking down during a fight with Dr. Peter Benton, Carter has agreed to go to rehab. The last scene of the episode shows him looking out of the window of a plane. The camera pulls back and we see that Dr. Benton is sitting next to him. If you watched the show and remember their often-contentious relationship and Benton's gruff demeanor, then you know what a wonderful thing this was. ER had a ton of amazing moments but, for me, this was the absolute best.
Clooney Connection: Duh.