Combine shallot, vinegar, and 1 tablespoon oil and set aside.
Preheat grill to medium-high. Drizzle remaining oil over chicken. Season with salt and pepper. Grill chicken until cooked through, 4 to 5 minutes per side. Let stand for 5 minutes.
Toss shallot mixture, cherries, and arugula in a small bowl. Season with salt and pepper. Serve on top of chicken.