Peel the potatoes and cut them into matchstick-sized pieces using the large shredding disk of a food processor, the julienne blade of a mandoline, or the large holes of a box grater. Blot any excess moisture with kitchen towel.
Film the bottom of an 11 or 12-inch nonstick skillet with about half of the clarified butter and heat for several minutes over a medium-high burner. Add half of the potatoes, sprinkle with salt and pepper, then add the balance and season again.
Using a large, sturdy spatula, press down on the potatoes to compress them into a pancake as they cook. Because you want the natural starch from the potatoes to hold the galette together, do this frequently as the potatoes brown, shaking the pan occasionally as well to keep it from sticking.
Once the first side is golden brown (about 3 minutes), reduce the heat to medium and cover the pan. Continue cooking until the potatoes on the top are tender, 6 to 7 minutes longer.
Now the galette needs to be flipped over. One easy way to do this is to slide it onto a baking sheet, the turn your pan upside down on top of it and flip both together.
Raise the heat to medium-high again and cook until the second side is golden brown and crispy, 2 to 3 minutes longer. The galette can be kept warm in the oven, but Julia recommends serving it immediately.