Make-ahead (at least a day if possible) danmuji: Mix water, vinegar, sugar, salt, garlic turmeric, peppercorns and bay leaves in a medium pot and bring to a boil, stirring until the the sugar is melted. Turn off the heat and add the radishes. Let cool to room temperature (1 to 2 hours). Transfer pickles and brine to an airtight glass container and store in the refrigerator. Pickles will last up to a month or so.

To make the sauce: Mix soy sauce, gochujang, garlic, ginger, vinegar, and brown sugar together in a small saucepan. Bring mixture to a boil over medium-high heat, reduce to a simmer, and cook until thickened enough to coat a spoon, about 5 to 7 minutes. Remove from heat, add 2 tablespoon sesame oil and let cool. Use immediately or store in an airtight container in the refrigerator.

For the Kimchi Mayo: add mayonnaise, kimchi brine, kimchi, sugar, sesame oil, garlic, and scallions to a bowl and mix until combined. Refrigerate until ready to use.

Form beef into 4 patties, about 1/2 inch wider than the burger buns. Press with your thumb to make a slight depression in the center to account for bulging as they cook. Season generously with salt and refrigerate until ready to cook.

Light one chimney full of charcoal. When all the charcoal is lit and covered with gray ash, pour out and arrange the coals on one side of the charcoal grate. Set cooking grate in place, cover grill, and allow to preheat for 5 minutes. Alternatively, set half the burners on a gas grill to the highest heat setting, cover, and preheat for 10 minutes. Clean and oil the grilling grate. Place burgers directly over hot coals, cover with vents open, and cook, turning occasionally, until burgers are well charred and the centers register 110°F on an instant-read thermometer for medium rare or 120°F for medium, 5 to 7 minutes total.

Brush burgers on both sides with bulgogi sauce and continue cooking, flipping and brushing occasionally, until they register 120°F for medium rare or 130°F for medium. Remove from grill and brush with more sauce on both sides until well glazed and coated. Sprinkle with sesame seeds.

Toast burger buns over grill until light golden brown. Spread bottom and top buns with kimchi mayonnaise. Divide half of cabbage over bottom buns. Top with cooked burgers. Brush with more bulgogi sauce. Top with remaining cabbage, close buns, and serve.