Preheat the grill to medium-high (400 degrees F), or get a grill pan hot over medium heat.
Combine chicken, ginger, garlic, half the green onions, and soy sauce.
Form mixture into 4 patties.
Rub burgers with oil.
Press a deep dimple into the center of each burger (this will help it stay flat while it cooks).
Mix kimchi, cabbage, 1 tablespoon Sriracha, vinegar, sugar, the remaining green onion, and 2 tablespoons mayonnaise together in a medium bowl; set aside.
In a separate bowl, mix remaining 2 tablespoons Sriracha and 2 tablespoons mayonnaise; set aside.
Grill burgers, covered, turning once halfway through, until cooked all the way, about 10 minutes total.
Assemble burgers, topping each with the slaw mixture and a dollop of the spicy mayonnaise.