Peel turnips and carrots, and cut them into large chunks. Cook them separately in cold, salted water, cover and bring to a boil. Simmer until very tender, 30-40 minutes (turnips) 20 minutes (carrots). Reserve the water to drink or use as broth.
Mash the chunks of carrots and turnips with a potato masher or fork (they will be slightly fibrous.)
Work in butter or oil with a generous pinch of nutmeg (optional), taste, and adjust the seasoning.
Add in milk if you desire smoother consistency.