Preheat oven to 350 degrees F (175 degrees C).
Whisk milk, eggs, vanilla extract, sugar, nutmeg, and salt together in a bowl.
Heat 1 1/2 teaspoons butter in a skillet over medium heat. Working in batches, dip each slice of brioche bread in the egg mixture and cook in the hot butter until golden, 2 to 3 minutes per side, adding more butter as needed. Transfer French toast to a baking sheet.
Bake in the preheated oven until puffed, 8 to 10 minutes.
Cook and stir 1/4 cup butter in a skillet over medium-low heat until browned and toasty, 3 to 5 minutes. Transfer browned butter to a cold dish. Drizzle browned butter over baked French toast and top each slice with 1 tablespoon maple syrup.
Challah bread can be used in place of the brioche, if desired.
Leave bread out overnight if fresh, in order to dry it out a bit.
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