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In a medium bowl, toss together radish, shallot, cucumber, cilantro, honey, lime juice, 1 tablespoon vegetable oil, garlic, salt and freshly ground pepper, to taste. Mash it all together a bit and let sit for at least 15 minutes.
Preheat oven to 475 degrees. Pat chicken dry and season generously on both sides with salt and pepper. Heat oil in a cast-iron or heavy nonstick skillet over high heat until hot but not smoking. Place chicken in the pan, skin side down; cook for 2 minutes. Reduce heat to medium-high; continue cooking, skin side down, occasionally moving the thighs and rotating pan to evenly distribute heat, until skin is golden brown, about 12 minutes.
Transfer skillet to oven and cook 13 more minutes. Flip chicken; continue cooking until skin crisps and meat is cooked through, about 5 minutes longer.
Serve chicken with cucumber radish relish.