Place the potatoes in a large pot, cover with cold water by about 1 inch, and add the 1 tablespoon kosher salt. Bring to a boil over high heat, reduce to a gentle boil, and cook until the potatoes are tender when pierced, 12β15 minutes.
Drain the potatoes thoroughly, return them to the hot pot, and let steam-dry for 2β3 minutes.
Meanwhile, heat the vegetable oil in a 10β12 inch nonstick or well-seasoned cast-iron skillet over medium heat. Add the onion and cook, stirring, until softened and lightly golden, 5β7 minutes.
Add the shredded savoy cabbage to the skillet and season lightly with a pinch of the kosher salt for seasoning. Cook, stirring, until wilted and tender with a few browned spots, 5β7 minutes. Transfer the cabbage-onion mixture to a bowl.
Roughly mash the hot potatoes with 1 tablespoon of the unsalted butter until coarsely smashed. Season with the remaining kosher salt for seasoning and the black pepper. Fold in the cabbage-onion mixture until evenly combined.
Melt 1 tablespoon unsalted butter in the skillet over medium heat. Add the potato mixture and press it into an even, compact cake about 1 inch thick. Cook without moving until the bottom is deeply browned and crisp, 8β12 minutes.
Slide the cake onto a plate. Add the remaining 1 tablespoon unsalted butter to the skillet, then invert the cake back into the skillet to brown the second side. Cook until crisp and heated through, 6β10 minutes, adjusting the heat to prevent scorching.
Let rest 5 minutes, then cut into wedges and serve hot. Traditionally it pairs well with fried eggs, leftover roast meats, or brown sauce.