Preheat your oven to 375°F (190°C). Using a sharp knife, carefully cut a pocket into each chicken breast. Pat the chicken dry with paper towels, then season both the outside and inside of each breast generously with salt, pepper, garlic powder, and Italian seasoning.
In a mixing bowl, combine the cubed mozzarella, halved cherry tomatoes, and chopped fresh basil. Toss gently to mix the ingredients evenly.
Carefully spoon the mozzarella and tomato mixture into each chicken breast pocket. Use toothpicks if needed to keep the filling inside during cooking.
Heat olive oil in an oven-safe skillet over medium-high heat. Add the stuffed chicken breasts and sear for about 3 to 4 minutes on each side, until golden brown.
Drizzle the balsamic vinegar over the chicken breasts, then transfer the skillet to the preheated oven. Bake for 20 to 25 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (74°C). The mozzarella should be melted, and the tomatoes soft.
Remove the skillet from the oven and let the chicken rest for 5 minutes. Remove toothpicks before serving. Optionally, drizzle with a balsamic glaze made by reducing balsamic vinegar on the stove.