Preheat oven to 400 degrees F.
Heat clarified butter in a 10 1/2-inch cast iron skillet over medium heat until very hot, but not smoking. Place 1 sprig of thyme in center of pan. Arrange the potatoes in an overlapping spiral to fill pan (be careful the fat may spatter). Season with salt and pepper. Brush bottom of a medium saute pan with clarified butter. Place over potatoes to weight 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 pan. Place a sprig of thyme in center of pan. Return galette to pan, 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 saute pan 5 minutes before finished baking. Carefully slide out onto cutting board. Cut into wedges and serve immediately.
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