Bring a large pot of well-salted water to a boil. Add the cabbage, return to a boil and cook until tender, 6 to 8 minutes. Drain well and set aside.
Melt 3 Tbs. of the butter in a 12-inch skillet over medium-high heat. Add the apples, spreading them evenly. Season with a generous pinch of salt, sprinkle with the sugar, and cook, tossing occasionally, until lightly caramelized and just tender (they should still offer a little resistance when pierced with a fork), about 5 minutes.
Add the wine and cook until reduced by half, about 1-1/2 minutes. Add the remaining 3 Tbs. butter, the cabbage, and caraway and continue cooking, stirring gently, until the butter has melted, 1 to 2 minutes. Season to taste with salt and a pinch more sugar, if necessary. Serve hot or warm.