Make the sauce: Heat the olive oil in a medium saucepan over medium heat until shimmering, about 1 minute. Add the sliced garlic and cook until fragrant but not browned, 30–60 seconds. Stir in the crushed tomatoes, red pepper flakes, 0.5 tsp kosher salt, and basil leaves. Simmer, stirring occasionally, until slightly thickened, 15–20 minutes; discard the basil and keep the sauce warm.
Heat the oven to 425°F. Line a rimmed baking sheet with foil and set a wire rack on top.
Prepare the chicken: If the breasts are thick, slice them horizontally to make 4 even cutlets, then pound to an even 0.5-inch thickness. Pat dry, then season both sides with 1 tsp kosher salt and 0.5 tsp black pepper.
Set up breading: Put the flour in a shallow dish. Beat the eggs in a second dish. In a third dish, combine the breadcrumbs, 0.5 cup of the Parmesan, dried oregano, garlic powder, 0.5 tsp kosher salt, and 0.5 tsp black pepper until evenly mixed, 30 seconds.
Bread the cutlets: Working one at a time, coat chicken in flour, shaking off excess, then dip in beaten eggs, then press into the breadcrumb mixture until fully coated. Set on the rack and let stand 5 minutes to help the coating adhere.
Fry: Heat the vegetable oil in a large skillet over medium-high until it reaches 350°F or a breadcrumb sizzles on contact, 4–5 minutes. Fry the cutlets in batches without crowding until deep golden and just cooked through, 2–3 minutes per side. Transfer back to the rack to drain for 2 minutes.
Assemble: Spread a thin layer (about 1 cup) of the sauce in a 9×13-inch baking dish. Nestle the fried cutlets on top. Spoon 2–3 tbsp sauce over each cutlet. Top evenly with the mozzarella, then sprinkle on the remaining Parmesan.
Bake until the cheese is melted and bubbling and the chicken registers 165°F, 10–12 minutes. For extra color, broil until browned in spots, 1–2 minutes.
Rest 5 minutes. Sprinkle with the parsley and serve hot with remaining sauce on the side.