Heat the oil in a medium saucepan, over medium heat. Add the carrots and saute for 2 minutes. Stir in the carrot juice and simmer until the liquid has reduced by 5/6. Put the carrot mixture in a blender and puree until smooth. Transfer to a serving bowl and serve at room temperature.
Mix all of the ingredients in a small bowl and let sit, ideally for 24 hours in the refrigerator. Serve cool.
Using the grating attachment in a food processor, grate the potatoes, onions, carrot and beet separately. Coat a small saute pan with a little oil and put it over low heat. Add half of the grated onions and saute them slowly until caramelized, soft and sweet. Using a colander, squeeze all the excess liquid out of the remaining vegetables and then combine all of the ingredients in a mixing bowl.
Add the 2 cups of oil to a medium skillet and heat it to 370 degrees F. Put 2 ounces of the latke mixture into the oil and cook until brown on 1 side, approximately 2 minutes. Using a slotted spoon or spatula, flip the latke and continue to cook until brown on the other side. Remove from oil to a paper towel covered tray and immediately season with salt, to taste. Continue with the remaining mixture. Arrange the latkes on a serving platter and serve immediately with the carrot puree and the horseradish and caraway creme fraiche.