In a large bowl, mix together the frozen hash browns, ham, red bell peppers, and onions. In a medium bowl, whisk together eggs, egg whites, condensed soup, milk, and black pepper. Pour over potato mixture and toss to mix. Add mixture to baking pan. Cover with foil and bake for 30 minutes. Remove foil, sprinkle with cheese and bake an additional 20 minutes until golden brown and cheese is melted.