Preheat oven to 375 degrees F. Coat 2-quart casserole with cooking spray.
To prepare custard: Whisk egg whites, eggs and milk in a medium bowl. Add mustard, pepper and rosemary: whisk to combine.
Toss bread, spinach, roasted red peppers and ham in a large bowl. Add the custard 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, about 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.