Heat a large skillet over medium-high heat. Add 1 tablespoon of the oil to the pan and swirl it around to coat. Season the chicken with ¼ teaspoon salt and pepper.
Add the chicken to the pan; cook for about 7 minutes per side, or until cooked through and golden brown. The meat should reach an internal temperature of 165°F. Remove the chicken from the skillet; set aside on a plate. Cover with foil to keep warm.
Do not clean out the skillet, but add another 1 ½ teaspoons of oil. Add the onion and rosemary; cook for 1 minute or until lightly golden, scraping up the bits from the bottom of the skillet as you stir.
Add the brown sugar, vinegar, mustard, ¼ teaspoon salt, and the chopped peaches. Reduce the heat so that the sauce is at a gentle simmer. Cook until the peaches soften and release their juices, about 10-12 minutes. Stir frequently, continually scraping the browned bits from the bottom of the pan to incorporate them into the sauce.
Return the chicken to the skillet; heat just until warm, spooning the sauce over the meat.