Pit and slice the peaches, set aside.
In a large nonstick pan, heat 1 tablespoon olive oil,
Season chicken breasts with salt and pepper, add to pan and sear on both sides
As chicken is searing, whisk together the remaining olive oil with vinegar, honey, mustard, and garlic. Season with salt and pepper, to taste
Once chicken is browned on both sides, add peaches to the pan, and continue to cook for 5 minutes, stirring once in a while.
When 5 minutes are up, add balsamic vinegar mixture and basil, stirring well to coat everything.
Flip chicken breasts at this point
Cook another 5 minutes or so, until chicken is cooked through, peaches are soft, and sauce has thickened and coated everything as a glaze.
Serve immediately