In a large skillet, heat the olive oil over medium heat. Add the diced onion and sauté for about 3 minutes until it becomes translucent.
Add the minced garlic and diced bell pepper to the skillet. Cook for an additional 2-3 minutes until the bell pepper softens.
Increase the heat to medium-high and add the ground beef. Cook until browned, breaking it apart with a spatula, about 5-7 minutes. Drain excess fat if necessary.
Stir in the soy sauce, hoisin sauce, sesame oil, ground ginger, and black pepper. Cook for another 2-3 minutes until everything is well combined and heated through.
To serve, spoon the beef mixture into the lettuce leaves. Top with shredded carrot and chopped green onions. Sprinkle with sesame seeds if desired.
Enjoy immediately while the filling is warm and the lettuce is crisp.