Preheat oven to 375F. Carefully butterfly each chicken breast by running a sharp knife horizontally through the thickest part of the breast but not all the way through. Both sides of the breast should lie flat on the surface. Sprinkle with salt and pepper.
To each chicken breast, add 1/4 of the bacon, 1/4 of the goat cheese, and 1/4 of the chopped basil leaves on one side of the breast. Fold the other half of the chicken breast over, then secure with a toothpick if desired.
When finished stuffing the chicken breasts, brown each side in an oven-safe skillet. Place skillet in oven and bake for about 25-30 minutes or until chicken reaches 165 degrees.
A few minutes before the chicken is done, prepare the balsamic glaze. Place the balsamic vinegar in a small saucepan over medium-high heat. Stir until the vinegar is reduced by 1/3. It will start to form a thick, syrup-like consistency.
To serve, place chicken breast on plate and drizzle glaze on top.