In a small bowl, whisk together the beef broth, soy sauce, brown sugar, cornstarch, and sesame oil. Set aside.
Brown the beef in a large wok or skillet over medium-high heat, breaking up the meat as it cooks. Add the onion, bell pepper, garlic, and ginger and cook for 3 to 5 minutes, until the vegetables soften. Season the beef and vegetables with salt and black pepper.
Stir the soy sauce mixture again and add it to the pan. Bring to a simmer. Reduce the heat to low and simmer for 5 minutes, until the sauce thickens.
Meanwhile, cook the ramen noodles in a pot of boiling water for 3 to 4 minutes (without the seasoning packet), or as directed on the package. Drain and transfer the noodles to individual shallow bowls. Spoon the beef mixture over the ramen and top with green onions, if using.