Boil potatoes in a large pot of water with a pinch of salt until tender, about 15 minutes. When cooked, a paring knife can be inserted into the centers and removed without resistance. Drain potatoes well and return to the hot pot. Let stand a few minutes to allow any remaining moisture to evaporate.
While potatoes are boiling, melt 4 tablespoons butter in a large skillet until foaming subsides. Add cabbage, 1 teaspoon kosher salt, and ½ teaspoon freshly-ground black pepper. Cook over medium-high heat, stirring frequently, until cabbage is just starting to brown, 5-8 minutes.
Add leeks and an additional ½ teaspoon of salt. Continue cooking, stirring often, until vegetables are tender, 5-8 minutes more. Add garlic, and cook for an additional minute. Stir in half and half, bring to a simmer, and remove from heat.
Stir vegetables and half and half into the cooked potatoes. Use a potato masher to fully combine and mash to desired consistency. Potatoes will thicken a bit as they stand. Season to taste with salt and pepper.
To serve, mound potatoes into a serving bowl, and use a spoon to create a shallow well in the center. Melt the remaining 2 tablespoons of butter and drizzle it over the potatoes, allowing it to pool into the well. Scoop potatoes, catching some of the melted butter with each serving. Garnish with snipped chives.
