Preheat the oven to 350°.
Wrap the beets in aluminum foil and place them into a baking dish. Bake until the beets just begin to soften, about 1 to 1 ½ hours. Remove the baking dish from the oven and cool the beets to room temperature. Remove the skin from the beets by rubbing with paper towels. Use a mandoline slicer to slice the beets as thin as possible. Place the beets into a bowl with red onion slices.
Whisk together the apple cider vinegar and granulated sugar in a separate bowl until the sugar is completely dissolved. Stir in the salt and pepper. Pour this liquid over the beets and onions and gently toss so that all the slices are coated in the brine. Set the bowl aside for at least 30 minutes. (You can cover and refrigerate the beets at this point for several days.)
Pour the balsamic vinegar and orange juice into a blender. Add the shallot and mustard. Pulse to emulsify. With the machine running, slowly drizzle in the olive oil. Season with additional salt and pepper. Place your serving platter into the freezer to chill. Increase the oven temperature to 400°.
Place the prosciutto slices onto a parchment line baking sheet. Bake until the prosciutto is crisp, about 5 to 8 minutes. Remove the baking sheet from the oven and cool the crisp prosciutto to room temperature. Cut the watermelon into ¼-inch slices. Use a 3-inch round biscuit cutter to cut medallions from the watermelon slices.
Use the mandolin slicer to thinly shave the fennel bulb. Remove your serving platter from the freezer and brush it with olive oil. Remove the beet slices from the brine. Lay the beets and watermelon onto the plater, slightly overlapping each round.
Break the prosciutto into pieces and scatter over the top. Scatter thin slices of fennel over all. Drizzle just a spoonful or two of vinaigrette over the carpaccio. Garnish with the fresh green leaves cut from the fennel.
