Preheat oven to 450°F. Line a baking sheet with foil and brush with 1/2 teaspoon oil; spread eggplant in a single layer on baking sheet and roast for 15 minutes. Remove from oven; pull up foil and seal edges to close. Let stand 15 minutes to allow eggplant to steam until tender.
Meanwhile, rub an 8-inch square baking dish with remaining 1/2 teaspoon oil. Combine marinara with 1/2 cup water. Spread 1/2 cup sauce mixture in baking dish. Place 2 noodles on top of sauce. You’ll need to break noodles to fit and form three rows, but don’t worry about making perfect pieces. Break off a third of each noodle to form the extra row.
Combine ricotta, goat cheese, 3 tablespoons basil, and pepper flakes. Dollop half of cheese mixture onto noodles, spreading carefully to cover. Top with half of eggplant slices and 3/4 cup sauce mixture. Repeat layers, beginning and ending with noodles each time.
Top noodles with remaining sauce mixture, spreading to cover edges. Cover baking dish with foil and bake at 450°F for 45 minutes or until noodles are tender and mixture is bubbly.
Uncover and top with Parmesan cheese and remaining basil; continue cooking 5 minutes or until cheese melts. Let stand 10 minutes before slicing and serving.