Preheat grill or grill pan over high heat.
Rinse and dry chicken with paper towels, then drizzle with olive oil and season with salt and pepper.
Grill chicken for 2 on each side. (Internal temperature should be 165°.)
Transfer to plate to rest and slice on diagonal.
In a medium jar add 1 cup cherries, vinegar, 1/4 cup olive oil, salt and pepper.
Shake well and refrigerate until ready to serve.
Toss together green leaf lettuce, radicchio, green onions, cucumbers and remaining cherries.
Shake dressing well and drizzle over salad.
Top with grilled chicken and garnish with parsley.