Place the unpeeled potatoes in a pot, cover with cold water by 1 inch, bring to a boil, then reduce to a gentle simmer and cook until just tender at the edges when pierced, 10–15 minutes.
Drain, let sit until surface moisture evaporates, about 30 minutes, then refrigerate uncovered until fully cold and dry, at least 8 hours or overnight.
Peel the chilled potatoes and coarsely grate them on the large holes of a box grater; fluff gently, then sprinkle evenly with the kosher salt and toss to distribute.
Heat 3 tbsp clarified butter in a 10-inch heavy skillet over medium heat until shimmering, 1–2 minutes.
Add the grated mixture to the skillet and pat into an even 0.75-inch layer, tucking the edges in to form a neat round. Cook uncovered for 5 minutes, then cover and cook 3–4 minutes to steam and bind.
Uncover and cook until the bottom is deep golden and crisp and the cake slides freely, 8–12 minutes, adjusting heat to avoid scorching.
Loosen the edges, slide the cake onto a plate browned-side down. Add the remaining fat to the skillet, then invert the cake back into the pan.
Cook until the second side is golden and the center is hot and steamy, 10–12 minutes.
Slide onto a plate, rest 2 minutes to set, then cut into wedges and serve.