Make the cabbage: In a large skillet, heat the EVOO over medium-high heat. Stir in the caraway seeds for 30 seconds. Add the cabbage and season with the sugar, vinegar, salt and pepper. Partially cover and cook, stirring frequently, until tender.
Make the gravy: In a medium skillet, melt the butter over medium heat. Whisk in the flour for 1 minute, then whisk in the stock, cider and worcestershire. Whisk in the gingersnap crumbs to thicken. Lower the heat and keep warm.
Make the burgers: In a large bowl, combine the beef, parsley, onion, relish and mustard. Form 4 patties, slightly thinner at the center and thicker at the edges for even cooking. In a large cast-iron or other heavy skillet, heat a drizzle of EVOO over medium-high heat. Add the burgers and cook, turning once, for 8 minutes. Transfer the burgers to the gravy and turn over to coat.
Place the burgers on the bread or roll bottoms and top with the cabbage and bread or roll tops. Serve with the chips.