Preheat oven to 375 degrees.
Butterfly the chicken by running a sharp knife horizontally through the thickest part of each breast. Place one hand on top of the breast to stabilize while holding the knife parallel to the work surface while cutting. Do not cut all the way through. Open the breast so both halves can lie flat.
Season the each chicken breast with salt and pepper, then add 1/4 each of the goat cheese, crumbled bacon, and sliced basil leaves to each breast.
Fold the breasts in half so the cheese mixture is surrounded by chicken breast on the bottom, back and top. Secure with long wooden toothpicks to help keep the stuffing in place, if desired.
Heat olive oil over medium-high heat in a large oven-safe skillet. Once hot, add the stuffed chicken breasts to the pan to brown. Once browned on the bottom, carefully turn each breast over and repeat on the other side.
Place skillet in preheated oven and cook for 230-35 minutes, depending on the thickness of your chicken breasts. Use a meat thermometer to check for doneness (chicken should be at 165 degrees F. Remove from the oven and cover until ready to serve.
A few minutes before removing the chicken from the oven, start making the balsamic glaze. Heat balsamic vinegar in a small sauce pan over medium-high heat. Stir while heating until the vinegar is reduced to about 1/3 of its original volume. It will form a thick, syrupy consistency when it is ready.
To serve, place a stuffed chicken breast on each plate and drizzle with balsamic glaze. Pairs well with a mixed green salad or roasted vegetables.