1. Place tofu on 4 layers of paper towels; top with a single layer of paper towel, another large cookie sheet and 2 large cans. Let stand 20 minutes.
2. Meanwhile, heat oven to 400 degrees F. Melt 3 teaspoons butter in a 12-inch nonstick skillet over medium heat. Add onion and cook 12 minutes, stirring occasionally. Sprinkle with thyme, increase heat to medium-high and cook 3 to 4 minutes more, until golden in spots and tender.
3. Toss potatoes, salt and pepper in a large bowl. Arrange half of the potato slices in a shallow 2-quart baking dish. Crumble half of the tofu over potatoes, then sprinkle with half of the onion and cheese. Repeat. Add broth and dot with remaining butter. Cover dish and bake 40 minutes. Uncover and bake 15 to 20 minutes more, until potatoes are tender. Let stand 10 minutes before serving.