Step 1Heat the oven to 350 degrees. Spread the chopped pecans on a baking sheet and place in the oven for 8 minutes, until lightly toasted. Remove from the oven and set aside.
Step 2Using a knife or vegetable peeler, remove the tough outer skin from the celery roots. Trim to square off the rounded edges so that each celery root is a 2 1/2 - or 3-inch cube. Use the julienne setting of a mandoline or a sharp knife to slice them into one-eighth-inch julienne. Transfer the celery root to a large bowl, add just enough cider vinegar to coat, then toss; this will keep them from oxidizing and browning.
Step 3Core the apples (unpeeled) and quarter them lengthwise. Cut each quarter crosswise into one-eighth-inch-thick slices. Toss them with the celery root, making sure they have just enough cider vinegar to coat.
Step 4Fold the currants and the pecans into the slaw.
Step 5In a small bowl, whisk together the sour cream, horseradish, Dijon mustard, sugar and 1 tablespoon plus one-half teaspoon cider vinegar. Season with 1 teaspoon salt and several grinds of pepper, or to taste. Gently fold the dressing into the slaw. Cover and refrigerate for at least 2 hours for the flavors to develop. The slaw will keep for up to 2 days, refrigerated.