In 10-inch nonstick skillet, melt butter over medium-high heat. Add chicken; cook 5 to 8 minutes, turning once, until chicken is browned on both sides.
Add apple juice. Reduce heat to medium; cover and cook 5 minutes. Add carrots; cover and cook 5 to 10 minutes or until juice of chicken is clear when center of thickest part is cut (170°F) and carrots are crisp-tender.
With slotted spoon, remove chicken and carrots from skillet; place on serving platter and cover to keep warm. Stir mustard into liquid in skillet. Spoon mustard sauce over chicken and carrots; sprinkle with peanuts.