Remove the stems and seeds from the bell peppers; set them aside. Melt the butter in a large skillet over medium low heat. Add the onions, celery, parsley, and garlic. Add the ground beef and saute together until the beef is completely cooked and the vegetables are lightly browned. Add the bread crumbs and season with salt and pepper. Remove from heat and let cool slightly.
Work the eggs into the stuffing. If more moisture is needed, add a little milk or broth to the stuffing mixture. Spoon the stuffing into the prepared bell peppers and place them in a shallow baking dish. Bake in a preheated 350° oven for 30 minutes or until the bell peppers are tender. Spoon tomato sauce or tomatoes over each stuffed pepper and sprinkle with shredded cheese; bake 8 to 10 minutes longer.