Whisk together soy sauce, brown sugar, water, hoisin sauce, rice vinegar, sesame oil, and cornstarch in a medium bowl until the sugar dissolves and cornstarch is fully incorporated. Set aside.
Bring a large pot of salted water to a boil. Cook noodles according to package directions until al dente (typically 6-8 minutes). Drain and rinse briefly with cold water to prevent sticking.
Heat vegetable oil in a large skillet or wok over medium-high heat. Add ground beef and cook until no longer pink, breaking it into small crumbles as it cooks (approximately 5-7 minutes).
Add minced garlic and grated ginger to the cooked beef and stir for 30 seconds until fragrant. Immediately add carrots and bell pepper, stir-frying for 2-3 minutes until vegetables begin to soften but still maintain a pleasant crunch.
Pour the prepared sauce into the skillet with the beef and vegetables. Bring to a gentle simmer and cook for 2-3 minutes until the sauce thickens slightly.
Add the cooked noodles and three-quarters of the sliced green onions, tossing everything together until the noodles are well-coated with sauce and heated through.
Transfer to serving plates or a large serving bowl. Garnish with remaining green onions, toasted sesame seeds, and red pepper flakes if desired. Serve immediately while piping hot.