Preheat oven to 350 degrees. Place beets in a roasting pan. Pour over 8 cups water, red-wine vinegar, sugar, and 3 tablespoons salt. Cover with parchment-paper-lined foil and roast until tender, about 40 minutes.
Drain beets and peel by rubbing with a kitchen towel. Cut beets in half lengthwise; set aside.
Juice 4 blood oranges into a small pot. Place over low heat and cook until reduced by half; let cool.
Transfer juice reduction in the jar of a blender, along with 3 red baby beets and 1/4 cup chardonnay vinegar; blend until smooth. With the motor running, drizzle in 1/4 cup olive oil and blend until emulsified; season with salt and pepper. Strain through a fine sieve into a medium bowl; set aside.
Slice the top and bottom of remaining oranges with a paring knife. Set one of the flat sides on your work surface. From top to bottom, following the curve of the fruit, cut away peel and white pith. Trim off any remaining pith. Holding the orange over a bowl, cut along both sides of each segment, staying close to the membrane, to release; set aside.
Toss the fennel slices, fronds, and arugula in a large bowl along with 2 tablespoons olive oil and 3 tablespoons chardonnay vinegar; season with salt and pepper. Divide evenly among 4 plates.
In a large bowl toss together remaining beets, orange segments, and half of the vinaigrette; season with salt and pepper. Evenly arrange on top of greens. Garnish with cheese and drizzle remaining vinaigrette around the perimeter of the salad.