Combine sugar, soy sauce, sesame oil and ground pepper. Stir until sugar dissolves. Add scallions, garlic and ginger into soy sauce mixture, stir. Combine this 'marinade' to the ground meats. Form meat into 6 patties. Let patties rest as you prepare the slaw.
For the slaw, squeeze excess liquid out of kimchee. Take the wringed ball of kimchee and thinly slice it (so it looks like cole slaw). In a bowl, combine kimchee with the sugar, sesame oil and sesame seeds. Set aside.
Grill patties about 5-6 minutes per side or until desired doneness. As patties rest after being grilled, prepare fried egg: In medium skillet, heat 2 tablespoons vegetable oil over medium-high heat. Crack 3 eggs, sunny side up (try to keep them separate). Carefully break yolk and then flip. Repeat with remaining 3 eggs and 2 tablespoons of oil.
Toast buns 1-2 minutes on remaining heat of grill. Place patty on bun, top with a fried egg and then finish it off with a spoonful of kimchee slaw.