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First, heat the grill to a medium heat. You can also use a grill pan if no grill is available. Once grill or grill pan is hot and just before grilling, oil the grill grate or grill pan.
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Next mix together the mayonnaise, peach juice, paprika, ¼ teaspoon of salt and a ¼ teaspoon pepper in a small bowl. Next, gently loosen the skin from the chicken with your fingers and dividing evenly, spread the mayonnaise mixture underneath. Be careful you don't want the skin to tear! Brush the chicken with 1 tablespoon of olive oil and place on grill with the skin-side facing down.
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Grill the chicken covered, until the skin is brown and crisp. An instant-read thermometer inserted in the thickest part of the meat should register 165° F. This is generally about 12 to 15 minutes per side.
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While the chicken is cooking, in a large bowl add in the peaches and cucumbers, with the shallot, thyme, vinegar, the remaining 2 tablespoons of oil, along with ¼ teaspoon each salt and pepper (or more to taste if you wish). Mix to coat all ingredients.
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Serve the chicken and the cucumber and peach salad.