Preheat oven to 400°. Place beets in a shallow baking dish.
Drizzle with 2 Tbsp. olive oil; season with salt and pepper. Add 1/2 cup water to dish, cover with foil, and roast until beets are tender (a paring knife inserted into center should meet no resistance), 40–50 minutes.
Let beets cool slightly, then peel skins. Meanwhile, stir sour cream and horseradish in a small bowl; season with salt and pepper; chill.
Whisk shallot, vinegar, mustard, and remaining 1/4 cup olive oil in a medium bowl; season marinade with salt and pepper. Add beets and toss to coat. Place flour in a shallow dish; season with salt and pepper.
Lightly beat eggs in a second shallow dish to blend. Place breadcrumbs in a third shallow dish; season with salt and pepper. Fit a large heavy pot with thermometer and pour in enough oil to measure 2".
Heat over medium-high until thermometer registers 350°. Working with a few at a time, remove beets from marinade, letting excess drip back into bowl. Dredge in flour, shaking off excess. Dip in eggs, letting excess drip back into bowl, then roll in breadcrumbs to coat. Fry beets, turning occasionally, until breading is golden brown all over, about 2 minutes.
Transfer to paper towels to drain; season with salt. Serve fried beets with reserved horseradish dip alongside.
Do Ahead: Beets can be marinated 2 days ahead. Cover and chill.