Gigot

Gigot (Gleason) is a mute Frenchman living in a cellar in the MÃ©nilmontant district of Paris in the 1920s. He ekes out a hand-to-mouth existence as a janitor at his landlady's apartment building. He is routinely treated with condescension by neighbors and often is made the butt of practical jokes. However, he is a decent and kindhearted fellow, traits not unnoticed by children and the animals he often feeds. Gigot has one unusual predilection: he is attracted by funeral processions and finds himself attending, whether or not he ever knew the departed. He can't help but cry along with all the other mourners.
After being abused by locals at a pub, he chances upon a woman, Colette (Katherine Kath), and her 6-year-old daughter Nicole (Diane Gardner), huddled in a doorway trying to stay dry. He takes them to his dingy basement abode, gives them what food and drink he has, a bed to sleep in, and shelter from the rain. Colette is suspicious but is so exhausted that she accepts.
Gigot gleefully dotes upon Nicole. Gigot is astonished to discover she is ignorant of what a church is, completely unaware of God. Nicole points to a crucifix and asks about it. The mute attempts to act out the story of Christ, beginning with Mary cradling the baby Jesus, from childhood through to the horror of crucifixion. Gigot rails at this Christ who has seemingly left him so inadequate. Nicole cries a single tear for him, then blows a reassuring kiss of love to the Christ upon the Cross.
Gigot entertains the little girl by dancing to his old gramophone, and by dressing as a waiter to feed his pet mouse. He is very protective, running alongside her on a merry-go-round to make sure she doesn't fall off. He also intervenes to protect Colette's honor while she is in the act of soliciting a john on a park bench near the merry-go-round. Gigot is trounced by the frustrated man for his trouble.
Furious over his interference with her "activities," Colette threatens to bolt with Nicole unless Gigot can provide her a life with a "man of means." Given only an hour to prove himself, Gigot happens past a bakery. The baker and his wife (having taken advantage of him for years) have been called away, thus leaving their till unattended. Gigot seizes the opportunity and steals their money.
With those ill-gotten gains, Gigot goes on a shopping spree, buying much-needed new clothes for Colette and Nicole, with a straw boater and a shave for himself. He buys a grand meal and drinks for all at a restaurant. But the good times are not to last â Colette's ex-boyfriend wants her back, and Colette succumbs. She expects to take Nicole along, but her pimp persuades her to wait.
The next morning, two bumbling bureaucrats try to remove Gigot to a home for the feeble-minded. Meanwhile, the baker has discovered the theft, and when Colette returns, Gigot and Nicole are missing. Gigot becomes a suspect, but he and Nicole are only playing at an abandoned basement chamber below the streets of Paris, while Gigot dances for her with so much gusto that the roof timbers fall in. They are nearly buried in rubble and Nicole is unconscious. Believing she is dead, Gigot rushes the girl to the church where the priest calls a doctor. Gigot runs into the angry mob, then flees into a coal barge loader and is washed into the river. He fails to resurface.
Thinking him dead, the locals are despondent over their despicable actions. In remorse they organise a funeral for Gigot, though all they have is his chapeau to bury. Gigot survived and is merely hiding. Unknowingly, he witnesses his own funeral procession and as usual is compelled to join it. When the time comes for the eulogy, he realizes it is for him. Gigot is spotted by the crowd and the chase begins again.

Question: What do they bury at Gigot's funeral?
Answer: chapeau; chapeau