Get out 2 rimmed baking sheets and set them aside. Take a large soup pot, add the whole, unpeeled potatoes and 1 1/2 tablespoons of the kosher salt. Add enough cold water to come 4 inches above the level of the potatoes and set over medium-high heat. Bring to a boil, then reduce the heat to maintain a gentle simmer, uncovered. Cook until the tip of a small, sharp knife can pierce the potatoes with no resistance, about 20 minutes. Drain the potatoes and gently spread them out in single layers on the rimmed baking sheets. Set them aside to cool at room temperature.
While the potatoes are cooking, cut the cabbage into slices 3/4 inch thick.
Melt half the butter in a large sauté pan over medium heat. Add the onions and the remaining 1 teaspoon of the measured salt. Sweat the onions until soft and nearly translucent but not browned, about 12 minutes.
Add the sliced cabbage to the onion and stir to combine. Cook until the cabbage is just starting to wilt, about 8 minutes (the cabbage should retain some crunch). Turn off the heat but leave the onion and cabbage in the pan.
When the potatoes are cool enough to handle, cut them in half.
Heat a large cast-iron skillet over medium-low heat and add 3 tablespoons of the oil. Add the halved potatoes—make sure they’re in a single layer, positioned so the sliced surface is face down against the skillet. Whatever you do, don’t crowd the pan—you want the potatoes to sear, not steam. (If you don’t own a skillet large enough, cook the potatoes in 2 batches.) Let the potatoes cook, undisturbed, until a nice caramelized crust forms on the cut surface, about 15 minutes.
Set the onion and cabbage, still in their skillet, over low heat, and add the browned potatoes. Using the flat side of a large meat tenderizer or a masher, roughly smash up the potatoes in the pan, barbarian style—they should be evenly broken up but still pretty chunky.
Cut the remaining butter into 4 or 5 slices. Add the slices to the pan 1 by 1, folding them in with a spatula or wooden spoon so they just melt into the potato mixture.
Turn off the heat and gently stir in the chopped parsley. Taste and adjust the seasoning with more salt if necessary.
Cook slowly, until the potatoes are well crisped and golden on the bottom, about 15 minutes (don’t be tempted to rush things by turning up the heat, or the onions will burn).
Carefully flip and crisp the other side (if the cake breaks up, no worries; just pat it back into shape with the spatula), about 10 minutes. Remove with the spatula or flip onto a warmed serving plate and serve immediately.