For the dip: Peel and roughly chop your horseradish root. Puree it in your food processor on high with the lemon juice, buttermilk, Dijon, and egg yolk. If the mixture is not a wet puree, add a few drops of water. Turn the machine to low and drizzle the oil until an emulsion is formed. Taste and adjust with salt, white pepper, and lemon juice, if necessary. Chill until ready to serve.
For the batter and pickles: Heat the peanut oil in your deep-fryer or Dutch oven to 375 degrees F.
Whisk together all the dry ingredients for your beer batter. Whisk together the wet ingredients, and then combine. Pat the pickle spears dry with a paper towel, dust with a little flour, coat with the beer batter, and gently drop into the hot oil. They are ready in about 2 minutes when they are golden brown. Season with salt and pepper. Serve with chilled Horseradish Buttermilk Dip.