Step 1Heat butter in large, wide pot over medium heat. Add onion and garlic and cook until onion softens, stirring often, about 5 minutes. Do not brown.
Step 2Add potato, parsley roots, celery root, 2/3 cup cream, milk, salt and pepper to taste. Cook mixture, uncovered, stirring often, until mixture is thick and vegetables are almost tender, about 25 minutes.
Step 3Transfer mixture to greased, shallow 6-cup baking dish. Smooth surface with spatula. (Can be made to this point a day ahead, cooled, then covered and refrigerated. Let come to room temperature before baking.) Spoon 2 tablespoons cream over mixture; spread with spatula.
Step 4Bake at 350 degrees until gratin bubbles and is lightly browned, 35 to 45 minutes. For a browner surface, place under moderate broiler 2 to 3 minutes. Serve hot.