Place the eggplant in a colander and sprinkle with 2 tablespoons salt. Set aside to drain for 30 minutes. Rinse well, drain, and pat dry with paper towels.

Lightly brush each slice with olive oil then grill over a medium-low flame. Turn once and baste with oil until golden brown. If using a broiler, cook the eggplant 12 inches from the heat source. Turn the slices once and baste often until they are brown. Remove and set aside.

In a large skillet, warm the vermouth over low heat. Add the onion and garlic, and sauté, stirring occasionally, until soft (about 5 minutes). Add the tomatoes, basil, salt, and pepper. Continue cooking until the mixture has thickened (about 10 minutes).

In a large pot, bring the water to a boil. Using a long-handled wooden spoon, slowly add 1 cup of the cornmeal, a little at a time, while stirring constantly. Reduce the heat to low and add the remaining cornmeal in a thin stream while stirring. Turn off the heat, stir the milk into the cornmeal, and set aside.

In a small bowl, combine the mozzarella cheese, cottage cheese, and egg. Set aside.

Preheat the oven to 350°F.

Lightly coat a 9 x 13-inch baking dish with vegetable oil. Spread half of the cornmeal mixture on the bottom. Top with the grilled eggplant slices, then sprinkle with 1-1/3 cups of the mozzarella mixture. Top with the tomato mixture and the remaining cornmeal. Using a spatula, spread the remaining mozzarella evenly on top.

Place in the middle of the oven and bake the polenta until it is golden brown (45 to 50 minutes). Wait 10 minutes before spooning the polenta into individual plates.