Peel the eggplants and cut them into large cubes. Place the cubes in a large saucepan and sprinkle them with 1 teaspoon of the salt. Add 2 cups water and bring to a simmer over medium-high heat. Reduce the heat to low, cover, and let cook until the eggplant is soft, about 45 minutes. Drain off the water and mash the eggplant well with a potato masher, or puree in a food processor or blender.
Preheat the oven to 350°F Brown the ground beef in a large heavy skillet over medium-high heat. Pour off the excess fat. Add the onions, bell pepper, celery, and garlic. Stir well, cover, and let cook until the vegetables are transparent, about 20 minutes. Stir in the thyme, the rest of the salt and the peppers, and cook for 5-7 minutes more. Mix in the pureed eggplant, cook 2 minutes more, and remove from heat. Stir in 1 3/4 cups of the bread crumbs.
Butter a casserole dish and sprinkle with the remaining bread crumbs. Pour in the eggplant mixture and bake in the oven until bubbling and just beginning to brown, about 20 minutes.