In large bowl, combine ground beef, garlic, ginger root, and pepper. Mix with a fork or your hands to blend. Do not overwork meat or your patties will become tough when grilled. Divide mixture into 4even-sized patties, about 1/2-inch thick. Place on plate and refrigerate. Grease barbecue grate and preheat grill to about 425 F.
In small bowl, combine mayonnaise, chili garlic sauce, and lime juice. Stir to blend. Set aside.
Place patties on preheated barbecue; grill for at least 3 minutes per side, flipping once, until meat thermometer inserted in thickest portion registers 160 F in center. Remove from heat to clean, warmed plate and cover to keep warm. Burgers will continue to cook as they rest.
Lightly spread cut sides of buns with some mayonnaise mixture. Place cut side down on grill and lightly toast.
To assemble, divide remaining mayonnaise mixture evenly among bottom halves of toasted buns. Top each with a couple of lettuce leaves. Place one patty on each. Top with equal amounts kimchi, shaved cucumber, and cilantro leaves. Add additional garnishes if you wish. Cover with top bun halves and serve.