Bring a large pot of heavily salted water to boil. Add turnips and cook for 25-30 minutes, until soft.
Meanwhile, in a medium saucepan, melt 1½ tablespoons butter over medium heat. Add garlic and leeks and cook, stirring occasionally, until they start to brown, about 10 minutes. Add salt and pepper to taste. Transfer to a bowl.
In the same pan, melt remaining ½ tablespoon butter. Add turnip greens and cook until they wilt, 2-4 minutes. Add salt and pepper to taste.
When turnips are soft, drain water and mash until no large chunks remain. Alternatively, you can pulse a couple times in a food processor. Stir in goat cheese and mix until well integrated. Add salt and pepper to taste.
To plate, place greens down first, top with mashed turnips and then leeks.