Combine 3 quarts water, potatoes, and 2 tablespoons salt in a large saucepan. Bring to a boil over high. Reduce heat to medium, and simmer until potatoes are fork-tender, about 25 minutes.
Meanwhile, thinly slice scallions, separating the white and light green parts from the dark green parts; set aside.
Melt ¼ cup butter in a large skillet over medium. Add garlic, caraway seeds, and white and light green parts of scallions; cook, stirring often, until fragrant, about 1 minute.
Add cabbage, pepper, and ½ teaspoon salt; cover and cook, stirring occasionally, until cabbage is tender, 20 to 25 minutes. (If cabbage starts to brown, add about ¼ cup water to skillet.) Remove from heat, and stir in vinegar.
Just before potatoes are finished cooking, microwave half-and-half in a medium-size microwavable bowl on high until just simmering, about 1 minute. Stir in remaining 6 tablespoons butter.
Drain potatoes in a colander, shaking off any excess water. Return potatoes to saucepan, and mash using a potato masher until smooth.
Stir in half-and-half mixture and remaining 2 teaspoons salt. Fold cabbage mixture into mashed potatoes until just combined.
Transfer mixture to a serving bowl, and top with additional pats of butter. Garnish with black pepper and reserved dark green parts of scallions. Serve immediately.
