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

Slice the sweet potatoes, parsnips, and beets into very thin rounds (about ⅛ inch thick).

Blend the sweet potato and parsnip slices with 4 tablespoons of heavy cream, sprinkle with Parmesan and thyme, and season with salt and pepper.

Coat the sliced beets with 2 tablespoons of cream and season similarly.

Pour ¼ cup of heavy cream into the bottom of the greased baking dish and sprinkle with Parmesan cheese and minced garlic.

Arrange the coated vegetables in the dish, alternating between sweet potatoes, parsnips, and beets.

Season the top layer with salt and pepper, then add remaining Parmesan cheese.

Cover the baking dish with aluminum foil and bake for 30 minutes.

Remove foil, sprinkle shredded Gruyère cheese over the top, and bake for another 18-20 minutes.

Allow to cool slightly and garnish with fresh thyme leaves before serving.