Preheat the oven to 350°F.
In a skillet over medium-high heat, brown ground beef, and drain excess fat. Variation: if you have some leftover roast, substitute shredded beef. Shredded chicken or finely chopped pork (as for stir fry) will work here too.
Reduce heat to medium, and stir in onions, mushrooms, parsley, garlic, red pepper, salt, and black pepper. Cook until onions are soft then remove from heat, and stir in cheddar cheese and ketchup. Variations: Try your favorite cheese, mozzarella, pepper jack or such — as long as it melts well.
Divide loaves of thawed bread dough into four equal pieces. On a lightly floured surface, roll each piece into a 6 inch circle. Variation: Wheat bread makes it a whole different sandwich.
Place about 1/2 cup of the beef mixture onto the center of each one. Carefully pinch together the edges to seal. Place buns seam side down on a lightly greased or nonstick cookie sheet.
Whisk together egg and milk with a fork. Brush the tops of the buns with the egg mixture and sprinkle with coarse salt and cracked black pepper.
Bake for 25 to 30 minutes in the preheated oven, until nicely browned.
Serve hot with condiments you like. Suggestions: sweet hot mustard, horseradish mustard, barbecue sauce, honey barbecue sauce, mustard barbecue sauce, sweet pickle relish, wasabe mayonnaise, or just pick your favorite!