In a bowl, combine the soy sauce, brown sugar, sesame oil, garlic, ginger, and black pepper. Whisk until the sugar is dissolved.
Add the sliced flank steak to the marinade, ensuring all pieces are coated. Cover and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Heat the vegetable oil in a large skillet or wok over medium-high heat.
Add the marinated beef to the skillet in a single layer. Cook for about 2-3 minutes, stirring occasionally, until the beef is browned and cooked through.
Add the sliced onion and carrot to the skillet and stir-fry for an additional 2-3 minutes until the vegetables are tender.
Remove from heat and stir in the chopped green onions and sesame seeds if using.
Serve hot over steamed rice or with lettuce leaves for wrapping.