Put the red beets in a small saucepan; cover with water. Repeat with the golden beets and the carrots. Cover each saucepan and bring to a boil. Reduce the heat to low and simmer until the vegetables are just tender, 15 to 20 minutes for the beets and 10 to 15 minutes for the carrots. Drain each vegetable and reserve separately. Set aside to cool. Quarter the beets. Place each vegetable including the radishes in its own nonreactive container.
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.