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

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

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

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 the minced garlic.

Stand the sweet potato slices upright on a bias in the baking dish, followed by a row of parsnips and then a row of beets. Repeat the layering with the remaining vegetable slices to create six rows.

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

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

Uncover the gratin and sprinkle the shredded Gruyere evenly on top. Bake uncovered for an additional 18 to 20 minutes, until fork tender and the top is slightly browned.

Remove from the oven and sprinkle with fresh thyme leaves before serving.