Step 1

Preheat oven to 350 degrees F (175 degrees C).

Step 2

Whisk milk, eggs, vanilla extract, sugar, nutmeg, and salt together in a bowl.

Step 3

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.

Step 4

Bake in the preheated oven until puffed, 8 to 10 minutes.

Step 5

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.