Prep your sauce. In a small bowl, whisk soy sauce, brown sugar, rice vinegar, gochugaru, and sesame oil. If using cornstarch, stir it into the sauce now.
Cook the aromatics. Heat a large skillet over medium-high.
Add a drizzle of oil, then the onion. Cook 2–3 minutes until softened. Stir in garlic and ginger for 30 seconds, just until fragrant.
Brown the beef. Add ground beef to the skillet.
Break it up with a spatula and cook until no longer pink and slightly crispy in spots, about 5–7 minutes. Drain excess fat if needed.
Sauce it up. Pour in the sauce. Stir and let it bubble for 1–2 minutes until glossy and slightly thickened.
Taste and adjust heat or sweetness. Remove from heat.
Prep the lettuce and toppings. Separate and pat dry lettuce leaves. Arrange green onions, carrot, cucumber, sesame seeds, and kimchi in small bowls for easy assembly.
Assemble the wraps. Spoon a bit of rice onto a lettuce leaf (optional), top with a scoop of beef, then add crunchy veggies, kimchi, and a sprinkle of sesame seeds.
Serve immediately. Enjoy while the beef is hot and the lettuce is crisp.