Preheat oven (electric cooker: 200°C/circulating air: 175°C/gas: level 3). Peel and halve onions. Clean, peel and halve the beetroot (beetroot stains strongly - it is best to wear disposable gloves).
Place both in an ovenproof dish with the cut surface facing down. Drizzle with oil. Bake in a hot oven for 50-60 minutes until everything is soft. Turn both after about 30 minutes.
In the meantime, peel and wash the potatoes, cover and cook in salted water for about 20 minutes. Grate cheese. Wash parsley, shake dry and chop. Warm the milk.
Drain the potatoes, add the milk and butter, mash everything. Season with salt and horseradish, stir in parsley. Spread the puree on the vegetables about 15 minutes before the end of the baking time. Sprinkle with cheese and bake until done.