Preheat the oven to 350°F. Wrap the red and golden beets in aluminum foil and place in a baking dish. Roast until tender, approximately 1 to 1.5 hours.
Cool the beets to room temperature, then remove skins using a paper towel. Utilizing a professional mandoline, slice the beets into paper-thin rounds and place in a bowl with sliced red onions.
Prepare the brine by whisking together apple cider vinegar, sugar, salt, and pepper until dissolved. Pour over the beets and onions, ensuring all slices are submerged. Let stand for 30 minutes.
Prepare the vinaigrette by blending white balsamic, orange juice, shallot, and mustard. With the blender running, slowly drizzle in the olive oil to create a stable emulsion. Season with salt and pepper.
Increase oven temperature to 400°F. Place prosciutto on a parchment-lined sheet and bake until crisp, approximately 5 to 8 minutes.
Cut the watermelon into ¼-inch slices. Use a 3-inch round biscuit cutter to extract medallions. Use the mandoline to thinly shave the fennel bulb.
To assemble, place a serving platter in the freezer for 10 minutes to chill. Remove beets from the brine. Arrange overlapping beet and watermelon medallions across the chilled platter.
Finish by scattering broken pieces of crisp prosciutto and shaved fennel over the rounds. Drizzle sparingly with the orange-balsamic vinaigrette and garnish with fennel fronds.