Sauté onion in butter in a large, heavy skillet over medium heat until soft, about 5 minutes. Add 1/2 teaspoon salt, 1/4 teaspoon pepper and nutmeg and continue cooking for one minute. Stir in spinach, remove from heat and set aside.
Spread one third of the bread cubes in a well-buttered 3-quart gratin dish or other ceramic baking dish. Top with one-third of spinach mixture and one-third of each cheese. Repeat layering twice with remaining bread, spinach and cheese.
Whisk eggs, milk, mustard and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper together in a large bowl and pour evenly over strata. Cover with plastic wrap and chill strata for eat least 8 hours or up to a day.
The next day, let it stand at room temperature for 30 minutes while preheating the oven to 350°F. Bake strata, uncovered, in middle of oven until puffed, golden brown, and cooked through, 45 to 55 minutes. Let stand 5 minutes before serving.