1Combine brown sugar, rice wine vinegar, fish sauce, soy sauce, toasted sesame oil, minced garlic, minced ginger, and gochujang in a saucepan. Bring to a simmer until thickened.
2Season ground beef with salt and pepper, mix, and divide into portions. Use a burger press to form the patties, indenting the center to stuff with torn mozzarella, then seal with more beef.
3Heat oil in a non-stick skillet over medium-high heat. Add the stuffed patties, press lightly, and cook for 3-4 minutes per side.
4Combine shredded carrots, shredded napa cabbage, kimchi, shredded scallions, salt, pepper, sesame seeds, mayonnaise, and a portion of the prepared sauce in a bowl and mix thoroughly.
5Remove patties from the pan and let them rest.
6Add fresh butter to the pan and toast the brioche buns cut side down.
7Assemble the burger by layering the bottom bun, a patty, a drizzle of the remaining sauce, and the kimchi slaw before topping with the bun lid.