For the scallion mayo: Heat a medium cast-iron pan over medium-high heat. Whisk together the peanut oil, soy sauce, sugar and garlic. Toss in the scallions to coat, and then add to the pan in a single layer. Cook until they soften and have just begun to blacken, about 30 seconds, then flip and char the other side, 30 seconds longer. Remove from the pan and let cool.
Finely chop the scallions and mix with the mayonnaise in a small bowl. Season with salt and pepper. Refrigerate for at least 30 minutes before using to allow the flavors to marry.
For the burgers: Preheat a grill or grill pan to medium-high heat. Divide the beef into twelve 2-ounce patties, 1/2-inch thick and 2 1/2 inches wide. Brush the patties with the oil and sprinkle both sides with salt and pepper. Grill the patties until the meat is visibly cooked more than halfway up the side, about 2 minutes, and then flip and cook until medium rare, 2 minutes longer. Remove and let rest for 5 minutes.
Place into the steamed buns and top each burger with 1 teaspoon scallion mayo and a rounded tablespoon of the kimchee. Serve.