Preheat the oven to 350 degrees F (175 degrees C).
Bring a large saucepan of water to a boil. Add eggplant and cook until tender, 10 to 15 minutes. Drain and pour eggplant into an 8-inch square baking dish.
Melt butter in a 3-quart saucepan on medium heat. Add flour and mix to create a paste. Add tomatoes, bell pepper, onion, brown sugar, and salt and cook over medium to medium-high heat. Stir occasionally until mixture bubbles and thickens, 5 to 10 minutes. Pour mixture over the eggplant in the baking dish.
Cover the vegetables in the baking dish with Cheddar cheese and bread crumbs.
Bake in the preheated oven until hot and bubbly, 30 to 40 minutes.