The frenchiest movie there is, is Amélie. If you like Paris and if you know Paris, you will enjoy this movie. It is a story about a young woman fairly charming but a little bit emotionally defective.
She works as a waitress in a small bistro, Les Deux Moulins. The story of this different woman is slow and unfolds the way children's books do.
As some themes just never seem to get older, this movie went for " the young woman who is looking for love".
Why do you need to see this movie? I would say this is more of a girl's movie though it is really peculiar and if you want to see something extremely french, you have to see this maybe even more parisian than french.
It is like looking at a work of art moving. To be honest, images are better than words to describe the esthetics and loveliness of this movie.
Overall, Amélie is an odd but especially shy young woman who is going to extraordinary length to find the love of her life in a playful way and sometimes weirdly. Around her revolves a little world of odd and interesting people who are caricatures not fallen very far from the tree.