Preheat oven to 375 degrees then coat a 2-qt baking dish lightly with cooking spray.
Place a medium saucepan over medium heat then add cream, butter, garlic, thyme, salt and pepper. Bring mixture to a boil then remove from heat. Let mixture rest while you prepare turnips. Trim root and stem end from each turnip then peel using a vegetable peeler. Use a sharp knife or a mandolin to slice each turnip into 1/8" thick slices and set aside.
Layer ¼ of the sliced turnips into the bottom of the prepared baking dish by shingling the slices then sprinkle a quarter of the cheese over the layer of turnips. Strain cream mixture to remove the garlic and thyme then pour ¼ of the cream evenly over the cheese. Repeat the process 3 more times with the remaining ingredients.
Cover dish with foil, place on a baking sheet then bake for 40 minutes. Remove foil then bake for an additional 20–25 minutes, until the top is golden and the turnips are tender. Test doneness by inserting a paring knife into the center of the gratin. The turnips will feel slightly firm but not crunchy. Let rest 10 minutes before serving.
If you’re not too keen on the heavy cream, you can substitute with half-and-half!