In a small bowl, whisk together bourbon, brown sugar, Dijon mustard, apple cider vinegar, soy sauce, and garlic. Set aside.
Trim the chicken. If using larger chicken breasts, either cut into smaller tenders or butterfly into thinner pieces. Use a meat tenderizer to flatten chicken tenders to the same thickness. Sprinkle with salt and pepper.
Heat the avocado oil in a large skillet over medium high heat. Once the oil is hot, add the chicken in a single layer. Brown on one side for 4 to 5 minutes. Turn the chicken over and brown on the other side for another 3 to 4 minutes. Transfer the chicken to a plate and tent with foil to keep warm. Check for an internal temperature of 165F.
Deglaze the pan with a few tablespoons of bourbon sauce, using a spoon to scrape up the browned bits.
In the same pan, drizzle the remaining ½ tablespoon of oil. Add the onion and thyme and saute 1 to 2 minutes. Pour the bourbon sauce into the pan. Cook for 4 to 5 minutes, or until the sauce is reduced by half and slightly thickened.
Add the peaches to the pan and cook for 3 to 4 minutes, or until slightly softened. Return the chicken back to the pan. Spoon sauce and peaches over the top of the chicken. Sprinkle with additional fresh thyme, if desired. Remove from heat and serve.