Preheat the oven to 350°F (175°C) and lightly spray a 9x13-inch baking dish with nonstick spray.
Trim the chicken breasts as needed and cut each large breast in half crosswise; then butterfly each piece by slicing nearly through the thickest side so it opens like a book.
Spoon about 2 heaping teaspoons of pesto onto the inside of each chicken half, then sprinkle approximately 2 heaping teaspoons of grated Parmesan over the pesto.
Fold the top half of each chicken over the filling to enclose it and place each stuffed breast into the prepared baking dish.
If any pesto or Parmesan remains, sprinkle it over the tops of the prepared chicken pieces.
In a saucepan over medium heat, combine the butter, cream cheese, minced garlic, and the 1/2 cup Parmesan for the sauce; heat until melted, whisking to smooth and remove lumps.
Pour the hot cheese sauce evenly over the stuffed chicken in the baking dish.
Bake, uncovered, for about 35 minutes or until the chicken is cooked through and reaches a safe internal temperature.
Remove from the oven, sprinkle with fresh basil chiffonade, drizzle with balsamic vinegar, and serve hot.