Peel, core, and cut the apples into ¼-inch-thick slices. Heat the butter in a saute pan over medium heat until it starts to foam. Add the apples and cook them until tender, about 10 minutes. Turn off the heat and stir in the brown sugar, cinnamon, and nutmeg. Set aside.
Lightly toast the bread. Cut the toast slices in half to make triangles. Butter a 9-by-13-inch baking dish, and arrange the bread in two rows, overlapping in the dish.
Beat the eggs in a medium bowl; then whisk in the sugar, milk, and vanilla to make a custard. Pour the custard over the bread triangles, and spoon the apples over the top. Cover, and refrigerate overnight.
The next day, heat the oven to 350°F.
Uncover the baking dish and bake for 50 to 60 minutes, until the custard is set and doesn’t shimmy when you shake the pan (err on the longer side to ensure its baked through–mine baked a little over 60 minutes total because I noticed a bit of runny custard bubbling out one of the sides around the 50 minute mark). It will puff up and brown slightly. Remove from the oven and let it rest for 10 minutes before serving. Serve it in squares or large spoonfuls with maple syrup, if desired.