In a small bowl, whisk together the soy sauce, hoisin sauce, rice vinegar, sesame oil, grated ginger, minced garlic, and cornstarch until smooth. Set aside.
Heat the vegetable oil in a large skillet or wok over medium-high heat. Add the ground beef and cook, breaking it up with a spoon, until browned and cooked through, about 5-7 minutes.
Add the diced carrot, red bell pepper, and the white parts of the green onions to the skillet. Stir-fry for 3-4 minutes until the vegetables are slightly softened but still crisp.
Pour the prepared sauce over the beef and vegetable mixture. Stir well to coat everything evenly. Cook for another 1-2 minutes until the sauce thickens and glazes the mixture. Remove from heat and stir in the green parts of the green onions.
Spoon the warm beef mixture into individual lettuce cups. Garnish generously with toasted sesame seeds. Serve immediately with optional sriracha or chili garlic sauce on the side.