Preheat the oven to 450 degrees F. Make dipping stations by adding flour to a plate or shallow dish. Whisk the eggs in another shallow dish. Combine breadcrumbs, parsley, and oregano in a third shallow dish. Set a wire rack on top of a baking sheet and set aside.
Place each chicken cutlet between two pieces of plastic wrap and pound with a meat mallet until they are an even thickness between ¼- and ½-inch. Season both sides of each chicken piece with 1 teaspoon of the salt and the black pepper.
Dredge the chicken in flour, shaking off any excess. Dip in the egg mixture, letting any excess drip off. Dredge in the breadcrumb mixture, pressing so it evenly coats both sides. Set aside and repeat with each piece of chicken.
Heat vegetable oil and butter in a large skillet over medium-high heat until hot. Working in batches if needed, add the breaded chicken to the hot oil. Cook for 2 to 3 minutes per side, flipping halfway through with tongs, until golden brown. You may need to decrease the heat to medium if the chicken is browning too quickly. Transfer the chicken to the prepared baking sheet with a wire rack. Lightly season with the remaining 1 teaspoon of salt.
Top each piece of chicken with ½ cup of the tomato sauce and 2 slices of mozzarella. Sprinkle with more parmesan. Bake in the preheated oven for 10 to 12 minutes until the cheese is melted, bubbling, and lightly browned in spots.
Meanwhile, cook the pasta in a large pot of salted, boiling water according to the package directions. Reserve 1 cup of the pasta water before draining.
Combine the remaining 2 cups of tomato sauce with the cooked spaghetti and ½ cup of the reserved pasta water. Toss to evenly coat the pasta, adding additional pasta water if needed.
Top the pasta with the chicken then sprinkle with parmesan and basil. Serve immediately.