Bread Eggplant: Cut two 3/4-inch planks lengthwise from center of eggplant, halve each plank crosswise. Cut remaining eggplant into 1/2-inch dice and set aside. Place flour in shallow dish. Beat egg in second shallow dish. Combine bread crumbs, 1/4-cup Parmesan, 1/4 teaspoon salt and 1/4 teaspoon pepper in third shallow dish. One at a time, coat eggplant slices lightly with flour, dip them in egg and dredge in bread crumb mixture, pressing to adhere. Transfer to wire rack set inside rimmed baking sheet and let sit 5 minutes (or refrigerate up to 1 hour).
Cook Eggplant: Adjust oven rack to upper-middle position and heat oven to 450°F Heat 1/2 cup oil in large nonstick skillet over medium-high heat until shimmering. Cook eggplant slices until lightly golden browned, about 1 minute per side. Transfer to wire rack set inside baking sheet and bake until eggplant is tender and deep golden brown, 12 to 15 minutes.
Make Sauce: Meanwhile, pour off oil and wipe out skillet with paper towels. Heat 2 tablespoons oil in empty skillet over medium-high heat until shimmering. Add the reserved chopped eggplant and 1/4 teaspoon salt and cook, stirring occasionally, until lightly browned, about 3 minutes. Stir in remaining 1 tablespoon oil, garlic and pepper flakes and cook until fragrant, about 30 seconds. Add tomatoes and reduce heat to medium. Simmer until eggplant is tender and sauce is thickened, about 5 minutes. Stir in basil and season with salt and pepper; cover and keep warm.
Assemble: Combine remaining 1/2 cup Parmesan and provolone in medium bowl. Top browned eggplant slices with cheese mixture and bake until cheese is melted, about 3 minutes. Transfer half of sauce to platter and top with eggplant slices. Spoon remaining sauce over eggplant. Serve.