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.