Lightly butter a deep 4-by-13 inch casserole dish. In a large bowl, whisk the eggs with the egg whites and sugar until smooth. Whisk in the heavy cream and vanilla extract.
Arrange the brioche in the casserole and pour the custard over the bread, making sure that each piece is submerged. Cover with foil and refrigerate overnight.
Preheat oven to 325°. Bake the bread pudding, covered, for 1 hour. Uncover and bake for about 25 minutes more, until the middle is set and the top is lightly browned. Let cool for 10 minutes and serve with maple syrup and stewed fruit or jam.