In a large bowl, combine the ground beef, ground pork, pepper, garlic powder, ground ginger, soy sauce, and breadcrumbs. Mix until well combined and form the mixture into 4 burger patties.
Heat a large skillet over medium-high heat. Add the kimchi and sesame oil then sauté for about 2 minutes. Set kimchi aside.
Next, cook the burger patties for about 4-5 minutes per side, or until they reach your desired level of doneness.
Evenly distribute the sautéed kimchi on top of the burger patties then top with cheese slices, if you're using. Cover pan with a lid for about 30 seconds to 1 minute until the cheese has melted.
In a small bowl, mix the mayonnaise and sriracha to make a spicy aioli.
Toast the burger buns lightly and spread spicy aioli on the bottom half of each bun. Top bun with burger patty then top with the other half of the bun. Enjoy immediately.