Preheat oven to 400˚F. Grease a 3-quart baking dish with butter.
Slice sweet potatoes, parsnips, and beets into very thin rounds (using a mandoline is easiest) and transfer each vegetable to its own bowl.
Pour 4 tablespoons cream over the sweet potatoes and parsnips, and 2 tablespoons cream over the beets. Top each bowl of sliced vegetables with ½ ounce grated Parmesan and 1 teaspoon minced thyme. Season each bowl generously with salt and pepper and toss together until all vegetable slices are well coated.
Pour remaining 1/4 cup cream into the bottom of a 3-quart (oval) baking dish and sprinkle with ½ ounce of Parmesan and minced garlic.
Grab a stack of sweet potatoes and line them standing up on a bias at the top of the baking dish. Follow the sweet potato with a row of parsnips, followed by a row of beets. Repeat with the remaining sweet potatoes, parsnips, and beets, creating 6 rows of root vegetables.
Season the top of the gratin with salt, pepper, and sprinkle the remaining Parmesan.
Cover with foil and bake for 30 minutes or until vegetables are soft.
Uncover the gratin and top with shredded gruyere.
Place the gratin back into the oven, uncovered, and continue to bake for an additional 18 to 20 minutes or until vegetables are fork-tender, the cheese has melted, and the top has lightly browned.
Finish with a sprinkle of fresh thyme leaves. Serve.