In a saucepan, combine cherries, red onion, basil, balsamic vinegar, honey & salt. Bring to a gentle simmer & cook for about 3-5 minutes. Stir in cornstarch/juice mixture & simmer until slightly thickened. Remove from heat & set aside.
Between 2 sheets of plastic wrap, gently pound chicken breasts to uniform thickness. Brush both sides of breasts with olive oil; season with salt & pepper. Grill on BBQ or in a saucepan on the stove until nicely browned on each side & cooked through. Serve with Cherry Chutney.