Yes, I thought of that scene too!

But it's an arrow, not a bullet ... and Clint is the removee, not the remover.
There's a scene with a candle flame and some doctoring in
Unforgiven, but again, it's not a bullet ... it's gashes from a bar fight, and again it's Clint who is the patient, while his companion, Morgan Freeman, is doing the doctoring, stitching up Clint's battered face.