Heat grill to medium-high. Brush the chicken with 1 teaspoon of the oil and season with ½ teaspoon each salt and pepper.
In a bowl, toss the onions, peaches, 1 tablespoon of the oil, and ¼ teaspoon each salt and pepper.
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.
Toss the arugula with the onions, peaches, vinegar, and remaining oil. Top with the cheese. Serve with the chicken.