Combine the spices, the vinegar, verjus, sugar and salt with 3 cups water in a medium saucepan. Bring to a boil. Pour the hot brine over each vegetable, dividing the brine equally. Let cool completely before covering. Refrigerate overnight and up to 2 days in advance. Serve the vegetables cold or at room temperature. They may be stored in the refrigerator for up to 2 weeks.
Heat the duck fat in a large saute pan. When the fat is hot, carefully place each duck leg in the pan, skin side down. Cook the legs over medium-high heat for 2 or 3 minutes on each side, or until lightly browned on both sides and heated through. Remove the legs from the pan and set aside on a clean paper towel to drain briefly before serving.
For the salad: In a small bowl, whisk together the sherry vinegar, Dijon mustard, and extra-virgin olive oil. Season, to taste, with salt and pepper. Toss the greens with the vinaigrette and drizzle with a few tablespoons walnut oil.
To serve place an equal amount of greens on each plate. Place a few pickled beets, carrots and radishes on each plate. Lay the warm duck leg over the greens, garnish with goat's milk cheese and serve at once.