Heat 1 tablespoon oil in a large skillet over medium-high heat. Reserve ¼ cup of the onion for garnish and add the rest to the skillet. Cook, stirring occasionally, until softened, about 5 minutes. Add the beef and cook until no longer pink, about 5 minutes. Drain off any fat. Add the Korean BBQ sauce and heat through. Keep warm until serving time.
Add 2 teaspoons oil to another skillet and heat over medium-high heat. Once the oil is hot, add the kimchi and sugar. Cook, stirring, until caramelized and heated through, 3-5 minutes.
Combine the mayonnaise with the Sriracha. Mix the reserved onion with the cilantro and sesame seeds. Spread the heated tortillas with a little of the mayonnaise mixture on each tortilla, top with a lettuce leaf, the beef, the kimchi, and finally the onion, cilantro, sesame seed mixture.