Preheat the oven to 350°F (175°C) and spray a 9x13-inch baking dish with nonstick cooking spray.
Trim the chicken breasts and cut each large breast in half lengthwise. If desired, butterfly each half by slicing almost through and opening like a book; leave about 1/2 inch uncut so it holds together.
Spoon about 2 heaping teaspoons of pesto onto each chicken half, spreading slightly.
Sprinkle about 2 heaping teaspoons of grated Parmesan on top of the pesto in each piece, then fold the top half over the bottom and place the stuffed chicken pieces in the prepared baking dish.
In a small saucepan over medium heat, combine the butter, cream cheese, minced garlic, and 1/2 cup grated Parmesan. Heat, whisking often, until the cheeses melt and the sauce is smooth and lump-free.
Pour the warm cheese sauce evenly over the arranged chicken. If any pesto or Parmesan remains, sprinkle it over the top.
Bake uncovered for 35 minutes, or until the chicken is cooked through and reaches an internal temperature of 165°F (74°C).
Remove from the oven, sprinkle with the chiffonade basil leaves, and drizzle with balsamic vinegar. Serve hot.
