GRIND MEAT Adjust oven rack to upper-middle position and heat oven to 375 degrees. Pulse onion and pepperoni in food processor until coarsely ground. Add beef and pulse until thoroughly combined.
MAKE SAUCE Transfer beef mixture to large saucepan and cook over medium heat, breaking up mixture with wooden spoon, until no longer pink, about 5 minutes. Using slotted spoon, transfer 1 cup beef mixture to paper towel-lined plate and reserve. Add tomato paste, garlic, and pepper flakes to pot with remaining meat mixture and cook until fragrant, about 1 minute. Stir in tomatoes and simmer until sauce is slightly thickened, about 20 minutes. Season with salt and pepper. (At this point, sauce can be refrigerated in airtight container for 3 days.)
PREPARE FILLING Combine ricotta, 2 cups mozzarella, 1 cup provolone, egg, salt, pepper, basil, and reserved meat mixture in large bowl. Pour 2 quarts boiling water into 13- by 9-inch baking dish. Soak noodles until pliable, about 5 minutes. Drain noodles on kitchen towel. Pour off water and dry baking dish.
ASSEMBLE DISH Spread half of meat sauce over bottom of baking dish. Following photos below, top each soaked noodle with 1/4 cup cheese filling, roll, and arrange, seam-side down, over sauce in baking dish. Spread remaining sauce over manicotti. Cover with foil and bake until bubbling around edges, about 40 minutes. Remove foil and sprinkle with remaining mozzarella and provolone. Bake until cheese is melted, about 5 minutes. Let cool 15 minutes. Serve.