Heat oven to 375°F.
Put beets in a baking dish and drizzle with olive oil. Add a pinch of salt and shake dish to coat. Fill dish with ¼ inch of water and cover with aluminum foil.
Wrap potatoes in foil and bake both potatoes (directly on oven rack) and beets for 50 to 60 minutes, until tender when pierced with a fork or paring knife. Remove from oven, discard foil, and set aside to cool.
Warm 1 tablespoon olive oil in a large skillet over medium heat. When warm, add onion and cook, stirring, for 2 to 3 minutes, or until tender. Transfer to a medium bowl. Wipe out skillet.
When beets are cool enough to handle, peel and finely dice. Add to bowl with onions.
Scoop potatoes from their skins and add flesh to bowl. Stir to combine. Add parsley, mustard, and cheese and stir to combine.
Add 1 tablespoon lemon juice and season with salt and pepper. Stir in egg, then add breadcrumbs, a few tablespoons at a time, until mixture holds together.
Form into 6 3-inch disks and place on a rimmed baking sheet. Cover with plastic wrap and refrigerate until firm, at least 20 minutes (can be made up to 2 days before serving).
Heat oven to 250°F.
Heat remaining 1 tablespoon olive oil in a skillet over medium heat. Cook fritters for 3 to 4 minutes per side, until crisp and browned. Transfer to a rimmed baking sheet and keep warm in oven until ready to serve.
In a small bowl, stir together crème fraîche, horseradish, remaining 1 tablespoon lemon juice, and salt to taste.
Serve fritters warm with a dollop of crème fraîche and a sprinkle of chopped dill.