Preheat your oven to 400°F. Grease a 3-quart (oval) baking dish with softened butter.

Slice the sweet potatoes, parsnips, and beets into very thin rounds. Transfer each vegetable to its own bowl.

Pour 4 tablespoons of heavy cream over the sweet potatoes, 4 tablespoons over the parsnips, and 2 tablespoons over the beets. Add ½ ounce grated Parmesan, 1 teaspoon minced thyme, and season with salt and pepper to each bowl. Toss gently to coat.

Pour the remaining ¼ cup (4 tablespoons) of cream into the bottom of the prepared baking dish. Sprinkle with ½ ounce of Parmesan cheese and minced garlic.

Layer the sweet potato slices upright on a bias, followed by a row of parsnips and then beets. Repeat until all vegetables are layered.

Season the top with additional salt, pepper, and sprinkle the remaining Parmesan cheese.

Cover the dish tightly with foil and bake for 30 minutes.

Uncover the gratin, sprinkle with shredded Gruyere, and bake for an additional 18 to 20 minutes.

Remove from the oven, sprinkle with fresh thyme leaves, and serve warm.