Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish with butter.
Step 2: Wash, peel, and thinly slice the sweet potatoes, parsnips, and beets into approximately 1/8-inch thick rounds. A mandoline slicer is highly recommended for uniform slices.
Step 3: In a medium saucepan, melt 1 tablespoon of butter over medium heat. Add minced garlic and sauté for 1 minute until fragrant. Do not let it brown.
Step 4: Pour in the heavy cream and vegetable broth. Add the fresh thyme leaves, a pinch of nutmeg, salt, and freshly ground black pepper. Bring to a gentle simmer, then reduce heat and let it cook for 3-5 minutes, allowing the flavors to meld and the sauce to thicken slightly.
Step 5: Arrange a single layer of mixed root vegetables at the bottom of the prepared baking dish. You can alternate colors and types for a beautiful mosaic pattern. Pour a portion of the cream sauce evenly over the vegetables, ensuring each piece is lightly coated.
Step 6: Repeat the layering process, alternating vegetables and sauce, until all the vegetables are used. Finish with a final generous pour of sauce. Top evenly with grated Parmesan cheese.
Step 7: Cover the baking dish tightly with aluminum foil and bake for 30 minutes. This helps steam the vegetables and ensures they become tender.
Step 8: Remove the foil and continue baking for another 20-25 minutes, or until the vegetables are fork-tender, the sauce is bubbly, and the top is golden brown and caramelized. If the cheese starts browning too quickly, you can loosely tent it with foil.
Step 9: Once baked, remove the gratin from the oven and let it rest for 10-15 minutes before serving. This allows the sauce to set and makes for cleaner slices. Serve warm as a stunning side dish.