Wash the beets and place them, unpeeled, in salted water with a little sugar, one tablespoon caraway seeds and a splash of vinegar. Cook until tender. Allow the beets to cool, then peel and dice them.
Peel the potatoes and cook them until done. Chop the peeled shallots and gherkins finely. Wash and finely chop the dill.
Put 2 tablespoons of the beets aside and place the rest into a pan with the milk, butter, beet juice, a little caraway and the orange peel and bring to the boil. Add the cooked, drained potatoes and coarsely mash the mixture with a potato masher.
Mix the oil, fruit vinegar, gherkins and diced shallots with the reserved beets and season with salt, pepper and dill. Serve this gherkin vinaigrette with the puree.