Place the beets in a small ovenproof pan with a little water; cover tightly with foil. Roast for 50 minutes to 1 hour, depending on the size of the beets. Test with a fork for tenderness. Cool, then cut off the stem ends and slip off the skins. Cut in half or in bite-size pieces; keep them warm until carrots are roasted.
Carrots will cook in less time than the beets. In a separate baking pan, coat them with the olive oil and roast for 15 to 25 minutes, depending on their size, until fork-tender. Combine beets and carrots with melted butter and juice; add salt and pepper and sprinkle with chopped dill.