Preheat the oven to 350°F (175°C) and spray a 9x13-inch baking dish with nonstick spray.
Trim the chicken breasts. Cut each large breast in half crosswise. Then butterfly each half by slicing nearly through the thickest side so it opens like a book, leaving about 1/2 inch uncut so it stays attached.
Spoon about 2 heaping teaspoons of pesto onto the inside of each butterflied chicken piece, then sprinkle roughly 2 teaspoons of grated Parmesan over the pesto.
Fold the top half of each chicken piece over the filling and place the stuffed breasts seam-side down in the prepared baking dish.
If you have any remaining pesto or Parmesan, sprinkle it over the arranged chicken in the baking dish.
In a saucepan over medium heat, combine the butter, cream cheese, minced garlic, and the remaining 1/2 cup grated Parmesan. Stir and whisk as it heats until the mixture melts into a smooth sauce and any lumps are gone.
Pour the warm 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 no longer pink in the center (internal temperature 165°F/74°C).
Remove from the oven, sprinkle with the chiffonade basil, and drizzle with balsamic vinegar. Serve hot.