Preheat the oven to 400 degrees. Heat clarified butter in a 10 1/2-inch cast-iron skillet over medium heat until very hot, but not smoking. Place a sprig of thyme in center of skillet. Arrange potatoes in an overlapping spiral to fill pan (be careful, because the fat may spatter). Season with salt and pepper. Brush the bottom of a medium saute pan with clarified butter, and place over potatoes to weigh them down. Cook until golden brown, about 8 minutes.
Carefully pour off excess butter, if there is any, and reserve. Turn galette out onto a baking sheet. Place another sprig of thyme in center of skillet. Using two spatulas, carefully return galette to skillet, browned side up. Season with salt and pepper. Cover with saute pan; transfer to oven, and bake until second side is brown and potatoes are cooked through, 20 to 30 minutes more. Remove top saute pan 5 minutes before removing skillet from oven. Carefully slide galette onto cutting board. Cut into wedges, and serve immediately.