Place potatoes, turnips, and parsnips in a large pot; cover with 2 inches cold water. Season water generously with salt. Bring to a boil, then reduce heat, and vigorously simmer until root vegetables are very tender and breaking apart, about 30 minutes. Drain.
Meanwhile, melt butter in a medium Dutch oven over medium heat. Add leeks, and cook until golden, about 15 minutes. Add root vegetables, and stir to dry out, about 2 minutes. Add milk and cream, and stir to break up vegetables. Heat until hot; season with salt and pepper.
Vegetables can be refrigerated 1 day. When reheating, add more milk to reach the desired creamy consistency.