Step 1Peel the turnips and cut them into quarter-inch by quarter-inch sticks. Place them in a container.
Step 2In a small saucepan, bring 1 1/4 cups water, the vinegar, 2 tablespoons salt, the sugar, red pepper, mustard seed, coriander seed, peppercorns and bay leaf to a simmer. Cook 10 minutes, then set aside to cool for 5 minutes.
Step 3Strain the hot liquid over the turnip sticks, seal tightly and refrigerate for at least 2 hours.
Step 4Cut a shallow cross-hatching on the skin side of the duck breasts, through the skin but not through the fat to the meat. Season the breasts liberally on both sides with salt and pepper, place on a plate, cover tightly with plastic wrap and refrigerate.
Step 5When ready to cook, remove the duck from the refrigerator to sit at room temperature for 15 minutes. Heat the oil in a nonstick pan over medium heat. When it is almost smoking, pat dry the skin side of the duck breasts with a paper towel and place the breasts skin-side down in the hot pan. Sear until the skin is a deep golden brown, 7 to 10 minutes. Turn the breasts and cook on the opposite side until they are medium-rare in the center, 3 to 5 minutes more.
Step 6Remove the breasts to a carving board; let rest for 10 minutes. Cut on a bias into crosswise slices about a half-inch thick. Divide the slices among the plates and garnish with some well-drained turnip pickles. Serve immediately.