Preheat oven to 375 degrees F. Coat an 11-by-7-inch glass baking dish or a 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, 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.
To wilt greens, rinse greens thoroughly in cool water. Transfer them to a large microwave-safe bowl. Cover with plastic wrap and punch several holes in the wrap. Microwave on high until wilted, 2 to 3 minutes. Squeeze out any excess moisture from the greens before adding them to the recipe.