12 (3/4"-thick) slices day-old white bread, such as Pullman, pain de mie, or country bread (a square-shaped loaf works best)
6 tablespoons unsalted butter
3 tablespoons all-purpose flour
3 cups whole milk
1/4 teaspoon freshly grated or ground nutmeg
1 1/4 teaspoons kosher salt, divided
1/2 teaspoon freshly ground black pepper, divided
3 1/4 cups grated Gruyère (about 10 ounces), divided
1/4 cup Dijon mustard
12 thin slices Black Forest ham (about 10 ounces)
6 large eggs
Nonstick vegetable oil spray