Wash eggplant; do not peel. Trim ends and cut eggplant crosswise into rounds 1/4" thick. Place on broiler pan or baking sheet, brush surface with oil, and broil about 3" from heat 5 min, until lightly browned. Turn, brush uncooked side with oil, and broil 3 min on other side. Place single layer of cooked eggplant slices on bottom of 2 qt. Baking dish (A deep casserole produces a thicker, creamier, multilayered dish; a shallow casserole gives fewer layers and a crustier top. Take your pick.) Cover with single layer of mozzarella cheese, spoon over enough sauce to cover, and sprinkle generously with grated Parmesan. Repeat layers until all eggplant is used, adding the Parmesan cheese layer every other time. End with tomato sauce and Parmesan cheese. Any extra sauce can be poured over top layer if needed for moisture and will filter down during baking. (At this point the casserole can be refrigerated for several hours if you like to work ahead. Remove from refrigerator and allow to come to room temperature before baking or increase baking time 5 to 10 min.) Bake in 350 F oven 15 to 20 min, until sauce bubbles and cheese melts. Serve from casserole with large cooking spoon, cutting cheese strands with scissors.