Melt 2 tablespoons butter in a skillet. Add the apples and brown sugar and cook for 7 to 9 minutes until the apples are just tender and the pan is nearly dry. Cool completely.
Butter a 9-by-13-inch dish. Layer the brioche and cooled apples evenly.
Whisk the eggs, milk, maple syrup, cinnamon, vanilla, nutmeg and salt, then pour over the bread. Press down gently.
Cover and refrigerate for 8 to 14 hours. Let stand out for 30 minutes while the oven heats to 350°F.
Bake uncovered for 50 to 55 minutes, until golden and set at 160°F in the center. Rest for 10 minutes before serving.