Shred the peeled carrots using the large holes of a box grater. Transfer to a medium bowl. Shred the peeled beet using the large holes of a box grater. Add 1/4 cup chopped dill to the bowl with the carrots and beets and toss to combine.
In a small bowl, use an immersion blender (or whisk) to combine 2 Tablespoons apple cider vinegar, 2 Tablespoons prepared horseradish, 1/2 teaspoon salt, and 1/2 teaspoon freshly ground black pepper. Slowly drizzle in 1/4 cup olive oil while continuing to blend, until the dressing is emulsified.
Pour the dressing over the shredded beet mixture and toss to combine. Serve with additional chopped dill, if desired.