Preheat the oven to 450°F. Combine the shallots, lime juice, mint, salt, pepper, and 4 tablespoons of the oil in a small bowl. Coat the chicken with 3 tablespoons of the vinaigrette, reserving the rest.
Heat the remaining oil in a large oven-proof skillet over medium-high. Cook the chicken skin-side down in batches in a single layer until the skin is golden brown and crisp, about 3 minutes. Turn, transfer to a baking sheet, and place in oven. Bake until an instant-read thermometer inserted in the center of a thigh registers 165°F, 15 to 20 minutes.
Meanwhile, toss the peach, cucumber, and watercress with 3 tablespoons of the vinaigrette. Divide the chicken and salad among 4 plates. Drizzle with the pan drippings. Serve with the remaining vinaigrette on the side.
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