Preheat the oven to 400°F (204°C).
Season the chicken breasts with 3/4 tsp salt and 1/2 tsp black pepper.
Cut a deep pocket into each chicken breast without cutting through, then stuff each pocket with 2–3 Tbsp shredded mozzarella and press the edges to close.
Set up three shallow bowls: one with the flour, one with the beaten eggs, and one with the bread crumbs mixed with the grated Parmesan.
Working one breast at a time, dredge the stuffed chicken in flour, shake off excess, dip into the beaten eggs, then coat evenly with the breadcrumb mixture.
Heat 3 Tbsp olive oil in a large deep skillet or Dutch oven over medium heat until shimmering, then add the breaded chicken in a single layer and cook about 4 minutes per side, until golden brown.
Pour the marinara sauce around the chicken and spoon a little sauce over each breast, then sprinkle the remaining mozzarella over the top.
Transfer the skillet to the preheated oven (or place chicken in an oven-safe dish) and bake uncovered 15–20 minutes, until an instant-read thermometer inserted into the thickest part of the chicken reads 165°F.
Remove from the oven and garnish with the chopped fresh basil before serving.