Root Vegetable Gratin

Name: Root Vegetable Gratin
Ingredients: 3 ounces finely grated Gruyère; 2 ounces finely grated Parmigiano-Reggiano; 2 cups heavy cream; 2 medium cloves garlic minced; 1 tablespoon fresh thyme leaves roughly chopped (plus more for garnish); Salt and black pepper; 4 pounds assorted root vegetables such as yams purple sweet potatoes and parsnips, peeled and sliced 1/8-inch thick on a mandoline; 1/2 pound butternut squash neck only, peeled and sliced 1/8-inch thick on a mandoline; 2 tablespoons unsalted butter