Preheat the oven to 400°F (205°C). Pat the chicken dry and season both sides and the inside of each breast pocket with 3/4 teaspoon salt and 1/2 teaspoon black pepper total, or to taste.
Cut a deep slit into each chicken breast to form a pocket, being careful not to cut through. Stuff each pocket with about 2–3 tablespoons of shredded mozzarella and press the edges closed to seal.
Set up a dredging station with three shallow bowls: flour in the first, beaten eggs in the second, and a mixture of bread crumbs and grated Parmesan (1/2 cup crumbs + 1/3 cup Parmesan) in the third.
Working with one stuffed breast at a time, dredge in flour and tap off excess, dip into the beaten eggs, then coat thoroughly with the breadcrumb-Parmesan mixture. Repeat for all breasts.
Heat 3 tablespoons olive oil in a large deep skillet or Dutch oven over medium heat. When hot, add the breaded chicken in a single layer and sauté until golden brown, about 4 minutes per side.
Pour 2 cups marinara sauce into the pan around and partly over the chicken; spoon some sauce onto each breast. Sprinkle the remaining mozzarella over the chicken.
Transfer the skillet to the preheated oven (or move chicken to an oven-safe baking dish) and bake uncovered 15–20 minutes, until an instant-read thermometer inserted into the thickest part of the chicken (avoiding the cheese pocket) reads 165°F (74°C).
Remove from oven, garnish with chopped fresh basil, and serve.