1. Heat grill to medium-high. Brush the chicken with 1 teaspoon of the oil and season with 1/2 teaspoon each salt and pepper. In a bowl, toss the onions, peaches, 1 tablespoon of the oil, and 1/4 teaspoon each salt and pepper.
2. Grill the chicken and onions until the chicken is cooked through and the onions are tender, 5 to 6 minutes per side. Grill the peaches until charred, 2 minutes per side.
3. Toss the arugula with the onions, peaches, vinegar, and remaining oil. Top with the cheese; serve with the chicken.