Prepare a grill for medium heat; lightly oil the grill grates with the vegetable oil. Pound the chicken breasts with a meat mallet between two sheets of plastic wrap to an even thickness of about 1/2 inch. Season generously with salt and pepper and the lemon zest. Grill the chicken until firm to the touch, 3 to 4 minutes per side.
Meanwhile, toss the peaches, tomatoes, basil, chives, lemon juice and olive oil in a medium bowl until well combined; season with salt and pepper.
Put a piece of chicken on each of 4 plates, and then divide the salad with the juices among the plates. Drizzle the chicken and salad lightly with olive oil.
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