Preheat oven to 450ºF (232ºC) with racks set in top and bottom thirds. Place a baking sheet on bottom rack while oven preheats. If serving with pasta, bring a large pot of water to a boil.
Combine Panko, Parmesan, oil, and ½ teaspoon each salt and pepper in a shallow dish. Lightly beat egg in a second shallow dish.
Halve chicken breasts horizontally into two thinner halves. Cover breast halves with plastic wrap and pound to an even ¼-inch thickness with a meat pounder; pat chicken dry.
Dip chicken pieces in egg, then dip in Panko mixture, pressing to coat thoroughly.
Working quickly, remove baking sheet from oven and coat with nonstick spray. Transfer breaded chicken to baking sheet and bake, on bottom rack, 6 minutes.
Remove sheet from oven, flip chicken pieces, then top each piece with 3 tablespoons mozzarella. Continue to bake chicken on bottom rack until mozzarella melts and an instant-read thermometer inserted into the thickest parts of chicken registers 160°F (71ºC), about 5 minutes.
Transfer sheet to top rack, turn oven to broil and broil chicken until cheese is golden brown and breading is crispy, 1–2 minutes.
Meanwhile, generously season boiling water with salt*, add spaghetti and cook according to package directions.
Drain spaghetti and return to now-empty pot.
Serve chicken over pasta with tomato sauce and parsley and/or basil.