Heat the vegetable oil in a large skillet over medium heat.
Add the finely chopped onion and cook for about 3 minutes or until translucent.
Stir in the minced garlic and ginger, cooking for an additional minute until fragrant.
Add the ground beef to the skillet, breaking it apart with a spatula. Cook until browned, about 5-7 minutes. Drain any excess fat if necessary.
Mix in the chopped bell pepper and shredded carrots, stirring until they are slightly softened, about 3-4 minutes.
In a small bowl, combine the soy sauce, hoisin sauce, rice vinegar, and sesame oil. Pour this mixture over the beef and vegetables, stirring to combine evenly. Cook for another 2-3 minutes.
Season with salt and pepper to taste. Remove from heat.
To serve, spoon the beef mixture into a lettuce leaf and top with optional toppings as desired.