In a large bowl, combine soy sauce, brown sugar, sesame oil, vegetable oil, garlic, ginger, black pepper, and red pepper flakes (if using). Whisk until the sugar is dissolved.
Add the sliced beef to the marinade, ensuring all pieces are well coated. Cover and refrigerate for at least 30 minutes, or up to 4 hours for more flavor.
Heat a large skillet or grill pan over medium-high heat. Add the marinated beef in a single layer, cooking for about 2-3 minutes on each side until browned and cooked through.
Add the sliced onion and carrot to the skillet, stirring to combine. Cook for an additional 2-3 minutes until the vegetables are tender.
Remove from heat and garnish with chopped green onions and sesame seeds before serving.