Thinly slice the turnips using a mandoline. Set aside.
For the custard: In a medium saucepan over medium low heat, combine cream, thyme, garlic, nutmeg, and salt and pepper. Bring to just below a simmer and cook until thickened slightly, 6 to 8 minutes.
To assemble: Preheat the oven to 375 degrees F.
In a 2-quart baking dish, add a thin layer of custard to cover the bottom. Then add a layer of turnips followed by a layer of cheese. Repeat the process of layering until all of the ingredients are used up, making 3 to 4 layers. Cover with foil and bake 45 minutes. Uncover and continue to bake until turnips are fork-tender with a golden crispy crust forms, about 15 minutes longer. Garnish with parsley and serve.