It's small, but the Friendship Moment the Professor and Luke share after musing over their likeliness to a particular bronze statue is absolutely adorable.
The entire sequence after Flora's abduction. Both Layton and Luke are very clearly worried sick; heck, even Don Paolo is anxious for her! The reunion scene in which she throws herself into a relieved Layton's arms elicited squeals of "Awww!" from more than one player.
Every single cutscene with Layton and Claire. That there was ever a person who could make him smile and blush like that is just amazing. Not to mention that she's the one who gave him his top hat and made him into the British gentleman we all know and love today. The look on his face after she gives him it is priceless.
Not only in that scene, but the mix between funny and heartwarming in the extra character profiles that are unlocked after you solve all of the puzzles of the game. If you tap on the picture of the professor as a young man once, he looks like we saw him in all of the flashbacks. But if you tap two times or so, the picture changes to him awkwardly wearing his newly-recieved top hat, looking quite puzzled, in love, and adorable.