Preheat oven to 375 degrees F. Coat a 7-by-11-inch glass baking dish or a 2-quart casserole with cooking spray.
Whisk eggs, egg whites and milk in a medium bowl. Add mustard, rosemary and pepper; whisk to combine. Toss spinach, bread, ham and roasted red peppers in a large bowl. Add the egg mixture and toss well to coat. Transfer to the prepared baking dish and push down to compact. Cover with foil.
Bake until the custard has set, 40 to 45 minutes. Uncover, sprinkle with cheese and continue baking until the pudding is puffed and golden on top, 15 to 20 minutes more. Transfer to a wire rack and cool for 15 to 20 minutes before serving.
Tip: To wilt spinach, rinse thoroughly with cool water. Transfer to a large microwave-safe bowl. Cover with plastic wrap and punch several holes in it. Microwave on High until wilted, 2 to 3 minutes. Squeeze out excess moisture before adding the spinach to the recipe.
MAKE AHEAD TIP: Prepare casserole through Step 2; refrigerate overnight. Let stand at room temperature while the oven preheats. Bake as directed in Step 3.