1. In a 1- to 1 1/2-quart pan over high heat, bring 1 1/2 cups cream and thyme to a boil. Remove from heat and add 1/2 teaspoon salt and 1/4 teaspoon pepper.
2. Peel sunchokes and potatoes, immersing in a bowl of water as peeled to prevent discoloration. If potatoes are wider than 1 1/2 inches, cut lengthwise into halves or quarters so they are closer to the width of sunchokes. Using a mandoline, food processor, or sharp knife, cut vegetables crosswise into 1/8-inch-thick slices. Immerse slices in water.
3. Drain vegetables well. In a large bowl, mix vegetables with 1/3 cup of the cheese. Scrape into a lightly buttered shallow 2 1/2- to 3-quart baking dish and spread level. Pour cream mixture over vegetables. Cover dish tightly with foil.
4. Bake in a 400º regular or convection oven until vegetables are very tender when pierced, 60 to 70 minutes. Remove foil. Spoon some of the cream in dish over vegetables; if there is very little liquid left in dish, drizzle an additional 1/4 cup cream over the top. Continue baking, uncovered, until vegetables are browned, 20 to 25 minutes longer.
5. Sprinkle remaining cheese evenly over vegetables. Bake until cheese begins to brown, about 5 minutes longer. Garnish with thyme sprigs. Season to taste with more salt and pepper.