Follow the steps in order for the best result.
Preheat your grill to high heat, aiming for a surface temperature between 450°F and 550°F.
In a large bowl, create the kimchi relish: Combine the shredded napa cabbage, 1/4 cup of the sliced green onions, rice vinegar, 1 teaspoon of minced garlic, 1 teaspoon of minced ginger, 2 teaspoons of toasted sesame oil, and 1/2 teaspoon of chili-garlic sauce. Mix well and set aside, stirring occasionally to allow the flavors to meld.
In a separate bowl, prepare the chili mayonnaise by mixing the mayonnaise with the remaining 1 teaspoon of chili-garlic sauce. Set aside.
Now, for the burger patties: In another bowl, gently combine the ground chuck, soy sauce, the remaining 1/4 cup of green onions, 1 tablespoon of minced garlic, 1 tablespoon of minced ginger, and 1 teaspoon of toasted sesame oil. Be careful not to overmix, as this can result in tough burgers. Shape the mixture into 4 patties, each approximately 1/2 inch thick.
Using tongs and a wad of paper towels lightly soaked in oil, carefully oil the grill grates to prevent sticking.
Grill the burger patties, covered, turning once halfway through, for a total of 7 to 8 minutes for medium doneness. During the last few minutes of grilling, toast the hamburger buns on the grill.
To assemble the Korean Kimchi Burgers: Spread the chili mayonnaise on the bottom halves of the toasted buns. Place a grilled burger patty on each bun bottom. Top each patty with a generous spoonful of the kimchi cabbage relish, using a slotted spoon to drain excess liquid.
Serve immediately with the remaining kimchi cabbage relish on the side.