Step 1Place the caraway seeds in a small saute pan and cook over medium heat, shaking occasionally, until they become aromatic and begin to pop, 3 to 5 minutes.
Step 2In a medium bowl, combine the creme fraiche with the toasted seeds and the horseradish. Season with one-half teaspoon salt and a pinch of pepper, or to taste. Cover and refrigerate for at least 24 hours and up to 3 days. Serve cool.
Step 1Heat the oil in a medium saucepan over medium-high heat until shimmering. Add the diced carrot and cook for 2 minutes, until the color is vibrant. Add the carrot juice and bring the mixture to a simmer, then reduce the heat to maintain a good simmer. Cook until the liquid has been reduced by a little more than half, 10 to 15 minutes, stirring occasionally.
Step 2Remove the mixture from the heat, cool for several minutes, and puree in a blender or with an immersion blender. Season with three-fourths teaspoon salt and a pinch of pepper, or to taste. Cover and refrigerate for up to a week. Serve at room temperature.
Step 1Heat the oven to 375 degrees. Trim the beet, wash it and wrap it in foil. Roast it in the oven for 1 hour, until a knife easily pierces through the flesh. Cool before handling; remove the skin.
Step 2Peel the potatoes, onions and carrot. Using a box grater, grate the potatoes, carrot and beet separately, squeezing the moisture out of each vegetable using a colander or paper towels, then combine the vegetables in a large bowl. Grate the onion, squeezing the moisture from half and adding it to a large bowl with the other vegetables. Reserve the remaining grated onion.
Step 3Heat a medium saute pan over medium heat. Add 2 tablespoons oil and the reserved grated onion. Saute the onion, stirring occasionally, until soft and caramelized, about 8 minutes. Remove from the heat and add to the bowl of grated vegetables, tossing to combine.
Step 4Stir in the eggs and flour, and season with one-half teaspoon salt and one-fourth teaspoon pepper.
Step 5Heat the remaining oil in a medium skillet until it shimmers. Drop the latke batter (beginning with a single "tester latke"), a quarter cup at a time, flattening slightly with a heat-proof spatula to form a pancake. Fry the latkes, 1 or 2 at a time, for about 2 minutes until they're a golden color on one side, then flip and cook 1 to 2 minutes on the other side. Remove the cooked latkes to drain on a paper towel-lined tray, sprinkling each with a good pinch of kosher salt, and continue with the remaining batter; you should have about 14 to 16 latkes.
Step 6Serve immediately with the horseradish-caraway creme fraiche and the carrot puree.