Preheat oven to 375° F. Coat a 7 x 11 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 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 - 20 minutes more. Transfer to a wire rack and cool for 15 - 20 minutes before serving.