Preheat oven to 375 degrees F (190 degrees C). Bring a large pot of salted water to a boil. Add potatoes and cook until tender but still firm, about 15 minutes. Drain, then mash with milk and butter until smooth.
In a large, deep skillet over medium high heat, combine ground beef, onion and green pepper. Cook until beef is evenly brown. Drain excess fat. Season with salt and pepper.
In a deep casserole dish, spoon a layer of beef mixture. Spread a layer of mashed potato over the beef. Continue alternating layers until the dish is full. Top with cheese.
