The Errand Boy (1961)
The Jerry Lewis talk with a doll in the end.
Tom: "Let's go rent 'The Errand Boy.'"
Movie scene from this Movie can be seen here
Jimmy mentions that the movie is included in the 50th anniversary boxset of The Nutty Professor (1963), and shows a clip of Jerry Lewis' character lip-synching to big band music
In a deleted scene from his episode, Peter imitates Jerry Lewis in one scene
In a deleted scene from the episode "Business Guy", shown in this episode, Peter imitates Jerry Lewis in one scene