**Cook the Noodles:** Bring a medium saucepan of salted water to a boil. Cook the noodles according to package instructions. Drain and set aside.
**Prepare the Sauce:** In a small mixing bowl, whisk together soy sauce, water, brown sugar, hoisin sauce, cornstarch, sesame oil, ginger, and garlic until smooth. Set aside.
**Cook the Ground Beef:** Heat vegetable oil in a large skillet or wok over medium-high heat. Add ground beef and cook until browned, breaking it apart with a spatula. Drain excess fat if necessary.
**Sauté Vegetables:** Add diced onions and shredded carrots to the skillet with the beef. Cook for 3-4 minutes, or until the vegetables are softened.
**Combine Sauce and Beef:** Pour the prepared sauce into the skillet with the beef mixture. Stir well and let the sauce simmer for 2-3 minutes, until thickened.
**Add Noodles:** Toss the cooked noodles into the skillet, mixing to coat evenly with the sauce. Cook for an additional 1-2 minutes until heated through.
**Garnish and Serve:** Sprinkle sliced green onions and sesame seeds on top. Serve immediately.