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. Spoon into a pastry bag with a large round tip; set aside. Preheat oven on broiler setting.
Heat a large heavy skillet over medium heat. Shape ground beef into 4 patties, and fry until evenly brown, about 5 minutes on each side. Fry the 4 eggs.
Place a burger on each piece of toast. Pipe mashed potato around the edge to form a wall on the edge of the toast. Heat the beans, and spoon onto each burger, inside the wall of potato. Cover with a slice of cheese.
Place burgers under broiler until cheese is melted, then top each with a fried egg.