Heat olive oil in a large skillet over medium high heat. Sprinkle both sides of the chicken with salt, pepper and dried rosemary. Add the chicken to the skillet and cook for approximately 7-8 minutes per side, or until cooked through and golden brown (time will vary depending on thickness of your chicken breasts).
Meanwhile, as the chicken cooks, bring a medium pot of water to a boil. Add the pearl onions and blanch for approximately 1-2 minutes. Transfer the onions to an ice bath to stop cooking and immediately remove from the ice bath. Trim off the root end and peel off the skin. Set peeled onions aside.
When chicken has finished cooking remove from the skillet and transfer to a clean plate. Cover with foil to keep warm.
In the same skillet add an additional teaspoon olive oil, diced onion, minced garlic and fresh rosemary. Stir to combine while scraping up the brown bits stuck on the bottom of the pan. Sauté for 1-2 minutes and add the white wine. Mix well to combine.
Add the reserved peach juice, brown sugar and dijon mustard to the skillet. Stir. Add the pearl onions and continue to cook over medium low heat for 3-4 minutes.
Return the chicken back to the skillet and add the peaches. Continue to cook for approximately 2-3 minutes, or until peaches are hot and chicken is warmed through. Season with salt and pepper to taste.
Serve with mashed potatoes, rice or buttered noodles.