Preheat a grill to medium-high. In a medium bowl, mix together the beef, lemon zest, ginger, half of the garlic, 1 tsp. sugar and 1 tsp. salt. Shape into four 4-inch patties; refrigerate while you prepare the slaw.
Meanwhile, in a large bowl, whisk together 1/4 cup EVOO, the lemon juice and remaining 1 tsp. sugar. Stir in the coleslaw mix and parsley; season with salt and pepper.
In a small skillet, heat the remaining 1/4 cup EVOO over medium heat. Add the remaining garlic and cook until golden, about 1 minute. Brush onto the cut side of each roll half.
Place the patties on the grill and cook, turning once, until firm around the edges, about 5 minutes for medium. Place the patties on the roll bottoms and top with some slaw, if desired, and the roll tops. Serve the remaining slaw on the side.
(Waste Not! Slice leftover fresh ginger and throw it into a jar with vinegar for a tasty addition to marinades and dressings.)