Using a sharp knife, cut a pocket into the side of each chicken breast (you'll later stuff cheese in it).
In a low, shallow bowl, add the flour. In another low shallow bowl, add the eggs and beat them. In a third bowl, add the panko breadcrumbs and Parmesan cheese. Stir to combine.
Dredge both sides of a chicken breast in the flour. Then, coat in the eggs. Lastly, dredge both sides in the panko mixture. Repeat with remaining chicken breasts. Sprinkle all chicken with salt and pepper.
Heat a large oven-safe skillet over medium heat. Once hot, add in the olive oil.
Once sizzling, add the chicken. Cook the chicken for 5 minutes on each side — the goal is not to cook it through but to just get some nice coloring.
Remove the chicken from the stovetop. Carefully, stuff each chicken breast with about 1-2 tablespoons of shredded mozzarella cheese.
Pour the red sauce over the chicken breasts. Bake in the preheated oven for 20 minutes.
Remove the chicken from the oven. Sprinkle with remaining shredded mozzarella cheese. Return to the oven to cook for an additional 5 minutes.
Once the cheese has melted, remove the chicken from the oven. Serve with cooked pasta.