For the Bulgogi Sauce: In a small saucepan, mix soy sauce, gochujang, garlic, ginger, vinegar and brown sugar together. Bring mixture to a boil over medium-high heat, reduce to a simmer, and cook until thickened enough to coat a spoon, 5 to 7 minutes. Remove from heat, add sesame oil and let cool. Use immediately or refrigerate in an airtight container for up to 1 week.
For the Kimchi Mayo: In a medium mixing bowl, stir together mayonnaise, kimchi brine, kimchi, sugar, sesame oil, garlic and scallions until well combined. Refrigerate until ready to use.
For the Burgers: Season with kosher salt the beef.Form beef into 4 patties, about 1/2 inch wider than the burger buns. Using your thumb, press a depression into the center of each patty to prevent bulging during cooking.
Light one chimney full of charcoal. When all the charcoal is lit and covered with gray ash, pour out and spread the coals evenly over half of coal grate. Alternatively, set half the burners of a gas grill to high heat. Set cooking grate in place, cover grill and allow to preheat for 5 minutes. Clean and oil the grilling grate.
Place patties directly over hot coals, cover with vents open and cook, turning occasionally. When patties are well charred and the centers register 110°F on an instant-read thermometer, brush each with the bulgogi sauce on both sides. Continue cooking burgers, flipping and brushing with sauce occasionally, until they register 120°F for medium-rare or 130°F for medium.
Remove patties from grill and brush with more sauce on both sides until well glazed and coated. Sprinkle all over with sesame seeds and let rest 5 minutes.
Meanwhile, toast burger buns over moderate heat on grill until light golden brown. Spread bottom and top buns with kimchi mayonnaise. Layer bottom buns with half the cabbage followed by the patties. Brush top side of patties once more with bulgogi sauce, then top with remaining cabbage, close top buns and serve.