Ombré Potato and Root Vegetable Gratin

Name: Ombré Potato and Root Vegetable Gratin
Ingredients: Unsalted butter, for greasing; 2 cups heavy cream; 3 garlic cloves, minced; 1 small shallot, minced; 1/2 teaspoon freshly grated nutmeg; 1 1/2 teaspoons kosher salt; 1/2 teaspoon pepper; 1 3/4 cups freshly grated Parmigiano-Reggiano cheese (5 ounces); 1 pound red beets, peeled and sliced on a mandoline 1/16 inch thick; 1 pound sweet potatoes or garnet yams, peeled and sliced on a mandoline 1/16 inch thick; 1 pound Yukon Gold potatoes, peeled and sliced on a mandoline 1/16 inch thick; 1 pound turnips, peeled and sliced on a mandoline 1/16 inch thick