For the Skewers:
Preheat the oven to 400°F. Wash the radishes, beets, and sunchokes in cold water. Dry thoroughly. Toss the beets in olive oil to coat generously and arrange in a single layer on a foil-lined baking pan. Wrap the top of the pan with more foil so the beets are covered and no moisture can escape. Put the beets in the oven and cook for 45 minutes. Remove to cool. Once the beets have cooled, peel them and slice into rounds. Using a ring mold, cut out circular pieces and set aside. Slice the watermelon and red radishes and cut out circular pieces; set aside. Alternate the vegetables and fresh parsley on soaked wooden skewers, folding the beet slices in half without breaking. Set aside.
For the Beet Vinaigrette:
Preheat oven to 400° F. Toss the beets in olive oil to coat generously. Put beets in a single layer on an aluminum foil-lined baking pan. Wrap pan with foil so beets are covered and no moisture can escape. Cook for 45 minutes. Once the beets are cooled, peel, trim, and cut into chunks; puree in a blender until smooth. Combine the vinegars, lemon juice, and water,. Add to the blender. While blending, slowly drizzle in the olive oil to form an emulsion. Mix until well combined and season with salt and pepper.
To Assemble and Serve:
Spoon some of the beet vinaigrette onto each serving plate and place two skewers on top of the vinaigrette. Lightly drizzle the skewers with olive oil and raspberry vinegar and sprinkle with the chives and fleur de sel.