Preheat the oven to 475°F.
Lightly grease a baking dish with cooking spray or oil.
Place the peeled and sliced turnips in a medium mixing bowl.
Add the olive oil to the bowl.
Pour the melted butter over the turnips.
Season with salt to taste.
Add the nutmeg.
Toss the turnips until they are evenly coated with the olive oil, butter, salt, and nutmeg.
Transfer the seasoned turnips to the prepared baking dish.
Spread the turnips into an even layer.
Sprinkle Parmesan cheese over the top, according to taste.
Place the fresh thyme sprigs over the turnips.
Bake uncovered for 15 to 20 minutes, or until the turnips are browned on top and fork-tender.
Remove the baking dish from the oven.
Discard the thyme stems before serving, if desired.
Serve warm.