Preheat about 1/4 cup of EVOO in a large, nonstick skillet over medium heat.
Beat the eggs with the milk or cream in a wide shallow dish and season with salt and pepper. In another shallow dish, combine the grated cheese with the bread crumbs and rosemary. Dip the eggplant slices in the egg mixture then coat them in the bread crumbs and cook in hot skillet for 3-4 minutes on each side. Work in two batches if necessary. Drain them on a paper towel-lined plate and reserve.
Remove the chopped eggplant from the oven and switch the broiler to high. Put the roasted pieces into a food processor and process them until they're smooth.
Preheat another large skillet over medium heat. Add in the remaining 2 tablespoons of EVOO and the garlic, red pepper flakes and the processed eggplant. Cook for about 1 minute then add tomatoes, parsley and some salt.
Toss the drained, hot pasta with half of the tomato-eggplant sauce and a ladleful of the pasta cooking liquid. Spread half of the pasta out in baking dish and top it with half of the breadcrumb-coated eggplant.
Add the remaining pasta to the baking dish and top it with the rest of the breadcrumb-coated eggplant. Pour remaining sauce over the top and sprinkle with provolone cheese. Place under the broiler for 2-3 minutes until the cheese is golden brown and bubbly.