Cut flank steak lengthwise into 1 1⁄2-inch-wide strips. Cut each strip across grain into slices that are 1⁄4 inch thick.
Mix 2 tablespoons vegetable oil, sesame oil, soy sauce, wine, 1 tablespoon garlic, 1⁄2 cup green onions, and black pepper in a large ziplock bag. Add beef and squeeze with hands to evenly distrib- ute mixture.. Refrigerate 6–8 hours.
Remove beef from marinade, transfer to a plate, and pat dry with paper towels. Reserve marinade.
Heat remaining oil in a wok or large skillet over high heat. Add ginger, remaining garlic, and
tablespoon green onions and cook until tender, 1–2 minutes. Add beef to wok and cook until medium-rare, 4–5 minutes. Remove beef from wok, set aside, and keep warm. Stir brown sugar into reserved marinade and pour into wok and bring to a boil. Reduce heat to medium-low and simmer to thicken marinade, about 5 minutes.
Assemble burritos by combining beans and rice then spooning onto tortillas. Top with beef and sprinkle with remaining green onions. Drizzle with sauce. Roll burritos as desired.