Wash the beets and place then in a saucepot large enough to contain the gallon of water, red wine vinegar, 1 cup of salt and 1 cup of sugar. Bring to a boil and turn down to a simmer, cooking until the beets are soft and a toothpick can be inserted easily into the beet. The beets will take approximately 1 to 1 1/2 hours to cook, depending on the size. If the water level begins to reduce below the beets, add more water to the pot until the beets are covered. When the beets are cooked, remove them from the water and allow to cool.
Peel the ginger and use a mandoline to slice 18 very thin pieces, slicing the ginger lengthwise. There should be a small piece of ginger left which should be set aside for later use. In a small saucepot heat the grape seed oil to 250° Fand add the ginger, fry until golden brown. Remove the ginger from the oil and place onto a paper towel, sprinkle with salt and set aside. Allow the oil to cool, it will have acquired a nice ginger flavor and you will use most of it in the other preparations for this dish. The rest you can refrigerate and use it for other dishes.
Place the peeled shallot and left over ginger into a blender with the mirin, Champagne vinegar, ½ teaspoon salt and the ¼ teaspoon sugar. While blending, slowly add a ¼ cup of the reserved ginger-flavored grape seed oil. When the oil is incorporated, check the seasoning, adjust as necessary and strain through a fine mesh strainer. Set aside.
Heat ¼ cup of the ginger-flavored grape seed oil in a medium sauté pan over moderate heat. Season the rock shrimp with salt and freshly ground pepper and add to the hot sauté pan. Cook the rock shrimp until just barely opaque in the center. Remove to a bowl and allow to cool. Once cool, marinate the rock shrimp with half the ginger vinaigrette. Set aside until ready to use.
Peel the beets and use the mandoline to slice into paper thin pieces. Arrange them directly onto 6 serving plates in a nice even layer. Brush the top of the beets with the remaining vinaigrette, and sprinkle lightly with salt and pepper. Divide the rock shrimp among the plates and place them on the beets. Thinly slice the radishes using a mandoline and divide them among the plates, arranging them in a scattered fashion. Finish by sprinkling the daikon sprouts on top of the beets.