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.
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.
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.
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.
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.