Preheat the broiler.

Wash and dry the eggplant and the zucchini; do not peel. Cut eggplant lengthwise into quarters, then cut it into 1/4-inch slices. Cut the zucchini into 1/4-inch slices as well.

Heat the oil in a large skillet over medium heat. Add the eggplant, zucchini, onion, and parsley and cook, covered, until tender, about 15 minutes. Stir in the mushrooms, reduce the heat to a simmer, and cook, uncovered, for 5 more minutes.

Meanwhile, in another large skillet, cook the ground beef over medium heat, breaking it up and stirring just until it is no longer red. Drain off the fat. Stir in the tomato paste, salt, oregano, and tabasco. Cook, stirring, until hot and well blended, 3 to 4 minutes longer.

Meanwhile, heat the milk in a medium-size bowl in the microwave for 2 minutes, or until hot. Add the grated cheese to the milk and stir until it melts, microwaving for an additional 15 seconds at a time if necessary.

Lightly grease a round, shallow 2-quart baking dish. Arrange the vegetables around the perimeter and put the meat mixture in the center. Pour the cheese sauce over everything. Broil for 2 or 3 minutes, until lightly browned.