Preheat oven to 350 degrees F.
Squeeze handfuls of spinach to remove as much liquid as possible, then finely chop.
Cook onion in butter in a large heavy skillet over moderate heat, stirring, until soft, 4 to 5 minutes. Add 1/2 teaspoon salt, 1/4 teaspoon pepper, and nutmeg and cook, stirring, 1 minute. Stir in spinach, then remove from heat.
In a large bowl, toss together the bread cubes, spinach mixture, 1 3/4 cups of the Gruyere, and 3/4 cup of the Parmigiano. Transfer 1/2 of the bread mixture to another bowl and toss with sausage.
Line a 9 by 13-inch baking dish with several pieces of foil, to form 2 enclosed sides, building a "dam" of foil down the center crosswise. Butter the foil. Put the "vegetarian" mixture in 1 side, and the "meat-eaters" mixture in the other.
Whisk together milk, eggs, mustard, remaining 1/2 teaspoon salt and 1/4 teaspoon pepper in a large bowl and pour evenly over each side. Sprinkle with the remaining 1/4 cup Gruyere, and 1/4 cup Parmigiano. Let the strata sit at room temperature for 30 minutes.
Bake strata, uncovered, in middle of oven until puffed, golden brown, and cooked through, 45 to 55 minutes. Let stand 5 minutes before serving.