Place the potatoes in a large pot with the 1 tablespoon of coarse salt and enough cold water to cover the potatoes by 1-inch. Bring to a boil, then reduce heat to a gentle simmer until the potatoes are just fork tender, 10 to 15 minutes. Drain the potatoes in a colander.
Return the pot to the stove and set over medium-high heat. Add in butter and once it's melted and bubbling, add in the cabbage. Sauté until the cabbage has softened and lightly browned, 6 to 8 minutes. Add the scallions and cook for 1 minute. Add in the garlic and cook until fragrant, 20 seconds. Season with salt and pepper, to taste.
Add the cooked potatoes to the pot along with the milk. Reduce heat to medium.
Using a potato masher, mash the potatoes until combined with the cabbage.
Taste and add in more salt and pepper, if necessary.
Transfer hot mashed potatoes to a serving bowl, make a slight well in the center and top with two thin slabs of butter. Garnish with a couple turns of fresh cracked black pepper and a sprinkle of diced scallions.
