Combine first 5 ingredients in a small saucepan. Season lightly with salt and pepper and bring to a boil over high heat. Reduce heat to low; simmer until peaches are very soft and flavors meld, about 10 minutes.

Remove pan from heat; let cool. Pour peach mixture into a blender and purée until smooth. Taste and adjust the salt and pepper to your liking.

Place half the sauce in a medium bowl; add the chicken and turn to coat. Let marinate at room temperature for 20 minutes, or cover and chill for up to 8 hours, turning occasionally. Cover and refrigerate remaining sauce.

Preheat grill to medium-high heat. Grill chicken until browned and almost cooked through, 4-5 minutes per side. Brush on all sides with reserved sauce; grill until glazed and cooked through, 1-2 minutes per side.

Slice the chicken against the grain and serve with the remaining sauce alongside.