Position a rack in the center of the oven and heat the oven to 425°F. Line a large rimmed baking sheet with foil and lightly coat the foil with nonstick cooking spray.
Mix the flour and 1/4 tsp. pepper in a wide, shallow dish. In a second wide, shallow dish, lightly beat the eggs with 1 Tbs. water. Put the panko in a third wide, shallow dish. Season the chicken with salt and coat each piece in the flour, tapping off the excess, then the egg, and then the panko, pressing the panko to help it adhere.
Heat 2 Tbs. of the oil in a 12-inch nonstick skillet over medium-high heat. Working in two batches, cook the chicken, flipping once, until the crumbs are golden and the chicken is almost cooked through, 1 to 2 minutes per side, adding 2 Tbs. more oil for the second batch. Transfer the chicken to the prepared baking sheet. Sprinkle the chicken with the Parmigiano and then top evenly with the mozzarella. Bake until the cheese is melted and the chicken is cooked through, 5 to 7 minutes.
Meanwhile, wipe the skillet clean and set over medium heat. Pour in the remaining 1 Tbs. oil and then add the onion and garlic. Cook, stirring often, until the onion is tender and lightly browned, 3 to 4 minutes. Stir in the tomatoes and 1/4 tsp. salt. Simmer, stirring occasionally, until thickened, 4 to 5 minutes. Remove from the heat and stir in the basil. Season to taste with salt and pepper. Serve the sauce over the chicken.