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

Carefully slice the sweet potatoes, parsnips, and beets into very thin rounds, ideally using a mandoline for uniform thickness.

In separate bowls, coat the sweet potato and parsnip rounds with 4 tablespoons of heavy cream each, and beet slices with 2 tablespoons. Add a portion of grated Parmesan and minced thyme to each bowl, mixing gently with salt and pepper.

Pour ¼ cup of heavy cream into the greased baking dish, spreading it evenly. Sprinkle with remaining grated Parmesan and add the minced garlic.

Layer the vegetables in the baking dish, standing them up vertically in rows, alternating between sweet potatoes, parsnips, and beets.

Season the top with additional salt and pepper, then sprinkle the remaining grated Parmesan evenly over the vegetables.

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

Remove the foil, sprinkle the shredded Gruyère over the top, and bake uncovered for an additional 18-20 minutes, until the cheese is melted and golden brown.

Let the gratin cool for a few minutes, garnish with fresh thyme leaves, and serve warm.