Game Plan: Slice and clean the leeks (see Cook's Note) while pan-searing the chicken.
Preheat the oven to 350 degrees F.
Heat a large skillet over medium-high heat; add the oil. Pat the chicken dry and season with some salt and pepper to taste. Lay the chicken skin side down in the skillet and cook undisturbed until the skin is golden and crispy, about 4 minutes. Add a nut-size bit of the butter to the skillet, and when the butter smells fragrant, turn the chicken and cook until opaque, about 4 minutes more. Put the chicken in a baking dish or roasting pan and bake just until firm to the touch, 10 to 12 minutes.
Pour all but 1 tablespoon fat from the skillet and return pan to the heat. Add the leeks and radishes and cook, tossing, until wilted, about 4 minutes. Season with the 1/2 teaspoon salt and pepper to taste. Add the vermouth, bring to a boil, and reduce until syrupy. Stir in the cream and bring to a boil. Remove the skillet from the heat, add the watercress, and toss just until it is wilted. Add the remaining butter and swirl the skillet until it melts. Stir in any chicken juices that have collected in the roasting pan. Spoon the leek mixture onto 4 plates and arrange the chicken breasts on top. Garnish with the reserved watercress sprigs and serve.
Quick-clean leeks by slicing in half lengthwise, then cutting crosswise. Place in a colander set into a big bowl of cool water. Swish sand and dirt free and lift out.
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