Preheat your oven to 400°F (200°C) and grease a 3-quart baking dish with unsalted butter.

Slice the sweet potatoes, parsnips, and beets into thin, even rounds using a mandoline slicer.

In separate bowls, mix sliced sweet potatoes, parsnips, and beets with heavy cream, Parmesan cheese, thyme, salt, and pepper.

Pour ¼ cup of heavy cream into the bottom of the greased baking dish and sprinkle half of the Parmesan and garlic over it.

Layer the sliced vegetables in the dish: sweet potatoes, then parsnips, then beets until all are used.

Season the top layer with salt and pepper and sprinkle the remaining Parmesan cheese on top. Cover with aluminum foil.

Bake covered for 30 minutes to soften the vegetables.

Remove the foil and sprinkle shredded Gruyère cheese over the top, then bake uncovered for another 18-20 minutes or until golden brown.

Let the gratin cool slightly before serving and garnish with freshly minced thyme.