Using a mandoline, cut the potatoes into matchstick sized pieces. Dry the potatoes with a cloth towel. As soon as the potatoes are cut and dry, film the frying pan with a 1/16 layer of clarified butter, and heat to very hot but not browning. Turn in the potatoes, making a layer about 3/8-inch thick. Sprinkle with salt and pepper, and 2 or 3 spoonfuls more butter, then press them down firmly all over with the spatula so they will flatten as they cook. Frequently press them down white they slowly brown on the bottom, and shake pan gently by its handle to be sure potatoes are not sticking to the pan. When browned, in 2 to 3 minutes, cover the pan and lower heat to moderate. Cook for 6 to 8 minutes, or until the potatoes are tender on top, but watch they do not burn on the bottom. Press them down again, and the galette is ready to brown on its other side. To turn it: either slide it out onto an oiled baking sheet, turn the frying pan upside down over it, and reverse the two so the galette drops into the pan, browned side up; or flip the galette in its pan, which, of course, is much more fun and faster.