For the eggplant: Place an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Spray a small baking sheet or a 9-by-13-inch glass or ceramic baking dish with vegetable oil cooking spray. Place the eggplant, cut-side up, on the prepared baking sheet. Make 1/2-inch-deep slits over the surface of the eggplant halves using a paring knife. Drizzle with the olive oil and bake until golden and tender, 40 to 45 minutes. Cool for 10 minutes.
For the filling: Scoop out 1 cup of flesh from the eggplant, using a spoon, and place in a food processor. Blend until the eggplant is smooth. Add the cheese, breadcrumbs, basil, egg, garlic and parsley. Pulse until the mixture is thick enough to form into balls, adding extra breadcrumbs as needed. Scoop 25 balls of the mixture, each about 2 tablespoons, onto a baking sheet using a small cookie scoop.
For the sauce: Bring the marinara and red pepper flakes to a simmer in a small saucepan over medium heat. Cook 5 minutes, stirring occasionally. Keep warm over low heat until ready to serve.
For assembly: Pour enough oil into a large heavy-bottomed saucepan to fill the pan about a third of the way. Heat the oil over medium heat until a deep-frying thermometer inserted into the oil reaches 350 degrees F. (If you don't have a thermometer, a cube of bread will brown in about 4 minutes.) In batches, fry the eggplant balls until golden, about 1 minute. Drain on paper towels.
Serve the fried eggplant warm or cold with marinara sauce and the grated pecorino cheese.
Recipe courtesy of Giada De Laurentiis