The Immortal Mr. Billy Joel
Two and a Half Men Season 8 Episode 5 (10/18/10)
Alan pretends to be Charlie at home while the latter is off getting plastic surgery. Alan manages to bring back a girl using his Charlie persona. It turns out the girl knew he wasn't actually Charlie, as she went home with Charlie before... but she likes to role play.
Charlie, meanwhile, pretends to go to Vegas but actually gets his eyes done at the surgeon.