In a shallow bowl, beat together eggs, half-and-half, sugar, vanilla, cinnamon, and nutmeg. Dip bread slices into egg mixture, coating both sides.
Preheat oven to 200 degrees F. Place a plate or serving platter in the oven. In a 12-inch skillet or on a griddle, melt 1 tablespoon of the butter over medium heat and spread it over the cooking surface. Add half the bread slices and cook for 4 to 6 minutes or until golden, turning once. Transfer slices to oven to keep warm. Repeat with remaining butter and bread. Serve with additional butter, maple syrup, and banana slices. Sprinkle with cinnamon.