Trim the beets at the top and bottom (remove any green leaves and snip off each ‘tail’). Put a large, double layer of foil in a roasting tin and set the beets in it, loosely pulling the foil up round them.
Drizzle on enough olive oil to moisten, season and sprinkle on 3 tbsp of the balsamic vinegar. Seal the foil at the edges of the package, creating a ‘tent’ around the vegetables (there needs to be room around them).
Bake until the beets are completely tender, which can take anything from 30 to 90 minutes, depending on size. Leave to cool, then slip off the skins and grate roughly.
Peel, halve and core the apples and dice the flesh. Melt 10g of the butter in a frying pan and sauté the apples, shallots and caraway seeds together for about 2 minutes.
Add the beetroot and sauté until everything is hot, then add the rest of the vinegar, the lemon juice and some seasoning. Taste for a sweet-savoury balance. Transfer to a warm serving dish.
Melt the rest of the butter in a small frying pan and add the paprika, heating until the butter is foaming. Spoon the sour cream on top of the beets and then pour the paprika butter over that. Serve immediately.