If the brioche is fresh, spread the cubes on a baking tray and dry them in the oven at 150°C (300°F) for 10 to 12 minutes, until they feel dry on the outside but are not browned. Let them cool. Dry bread soaks up the custard without falling apart; fresh soft bread turns to mush.
Brown the butter in a small pan over medium heat until it turns golden and smells nutty, about 3 minutes. Pour it into a bowl and let it cool for 5 minutes.
Whisk the eggnog, eggs, brown sugar, vanilla, nutmeg, cinnamon, salt, and the cooled brown butter in a large bowl until smooth.
Put the brioche cubes in a buttered baking dish, about 20 x 30 cm. Pour the eggnog custard evenly over the top. Press the cubes down gently with a spoon so they all soak. Cover the dish and refrigerate overnight, or for at least 4 hours.
Heat the oven to 175°C (350°F). Make the crumble: rub the flour, brown sugar, cinnamon, and cold butter together with your fingertips until the mixture looks like coarse crumbs with some pea-sized pieces left.
Take the dish out of the fridge and scatter the crumble evenly over the top. Cover the dish loosely with foil and bake for 30 minutes. Then take the foil off and bake for 20 to 25 minutes more, until the top is golden and crisp and the centre is set. Push a knife into the middle — it should come out with no runny custard on it. If the crumble starts to colour too fast, put the foil back on loosely.
Leave the bake to rest for 10 minutes before serving, so it slices cleanly. Dust with icing sugar and serve warm with maple syrup.
