Preheat oven to 400°F with rack in middle position. Place rutabaga, potatoes, celery root, cream, and thyme sprigs in a large saucepan. Cover and cook over medium, turning vegetables occasionally, until cream just begins to bubble around edges of pan. Continue cooking, turning vegetables occasionally, until vegetables are tender on the outside but still crunchy in the center, 12 to 15 minutes, reducing heat as needed to maintain a very gentle simmer. Remove from heat; stir in salt and pepper.

While vegetable mixture cooks, lightly grease a 2-quart baking dish or 10-inch cast-iron skillet with butter, and place on a rimmed baking sheet.

Remove and discard thyme sprigs from vegetable mixture. Spoon half of vegetable mixture into prepared pan; sprinkle with 2 tablespoons cheese. Top with remaining vegetable mixture, pressing down gently to even top layer. Sprinkle with remaining 2 tablespoons cheese. 

 Place baking sheet with gratin in preheated oven. Bake until mixture is bubbling and vegetables are tender, about 15 minutes. Increase oven temperature to broil (do not remove gratin from oven). Broil on middle rack until golden brown in spots, 1 to 2 minutes. Remove from oven; garnish with thyme leaves. Let cool 15 minutes.