Preheat the oven to 375 degrees F (190 degrees C). Lightly butter a 9-inch gratin dish.
Using a mandoline, slice potatoes and turnip into 1/8-inch-thick slices.
Layer 1/2 of the potatoes and turnips in the bottom of the gratin dish. Sprinkle 1/2 of the thyme leaves over the vegetables, season with salt and pepper, and top with 1 cup shredded Monterey Jack cheese.
Layer remaining 1/2 of the potatoes and turnips on top of the cheese. Sprinkle remaining thyme leaves over the vegetables. Season with salt and pepper again. Pour half-and-half evenly over the top and cover with remaining 1 cup Monterey Jack cheese.
Cover and bake in the preheated oven for 40 minutes. Uncover and continue baking until browned and bubbly, about 10 more minutes. Remove from the oven and let sit 10 minutes before serving.