How to Make It
Heat grill or a grill pan to medium. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate or grill pan.
Mix together the mayonnaise, paprika, and ¼ teaspoon each salt and pepper in a small bowl. Loosen the skin from the chicken with your fingers and, dividing evenly, spread the mayonnaise mixture underneath. Brush the chicken with 1 tablespoon of the oil and place on grill, skin-side down. Grill, covered, until the skin is browned and crisp and an instant-read thermometer inserted in the thickest part of the meat registers 165° F, 12 to 15 minutes per side.
Meanwhile, toss the peaches with the cucumbers, shallot, vinegar, thyme, the remaining 2 tablespoons of oil, and ¼ teaspoon each salt and pepper in a large bowl.
Serve the chicken with the peach and cucumber salad.