Combine the turnips, milk, thyme and garlic in a medium saucepan. Set over medium heat and partially cover the pan. Bring to a gentle simmer and cook for 20 to 30 minutes, until the turnips are tender-the tip of a paring knife should go through without resistance.
Drain the turnips, reserving the cooking liquid, and transfer to a food processor (discard the time sprigs). Add about 1 cup of the reserved cooking liquid and the butter, season with plenty of salt and pepper and puree until smooth. Add more of the liquid, if necessary. Serve hot.