Put bread crumbs in a bowl with enough water to cover; set aside to soak.
Heat vegetable oil in a large skillet. Cook and stir ground beef, onion, and garlic in hot oil until the beef is completely browned and no large chunks remain, 7 to 10 minutes.
Stir eggplant into ground beef mixture; cook and stir until eggplant is tender, about 5 minutes. Remove skillet from heat and set aside to allow mixture to cool.
Drain excess water from bread crumbs and dump into a large bowl; add ground beef mixture, half the tomatoes, rice, Italian cheese blend, egg, and basil. Stir the mixture until ingredients are evenly mixed, adding more liquid from the tomatoes if too dry; season with salt and pepper. Fill the eggplant shells with the mixture and put into a large baking dish. Top with remaining tomatoes and cover dish with aluminum foil.
Bake in preheated oven for 20 minutes, remove foil, and continue baking until beginning to brown around the edges, about 10 minutes more.