1. Generously butter a 5-cup gratin dish. Melt 1 tablespoon butter and olive oil in a large skillet over low heat. Add onions, sprinkle with sugar, and cook, stirring occasionally, until onions are very soft and golden, 15 to 20 minutes, and set aside.

2. Preheat oven to 400 degrees. Using a mandoline or a knife, slice potatoes 1/8 inch thick. Arrange one-third of the potatoes so they overlap in the gratin dish; sprinkle with salt and pepper. Cover potatoes with half of the reserved onions, half the cheese, and one-third of the thyme leaves. Arrange another one-third of the potatoes so that they overlap; sprinkle with salt and pepper. Repeat layering with remaining onions, cheese, and one third of the thyme. Arrange remaining potatoes on top so they overlap. Melt remaining 2 tablespoons butter; brush it over the final layer of potatoes. Sprinkle the remaining thyme, and season with salt and pepper.

3. Cover tightly with aluminum foil, and place in the oven. Bake for 40 minutes; remove foil, and bake until golden brown, 20 to 30 minutes more. Remove from oven, let stand 5 minutes, and serve.