Cook the noodles according to the package directions until just al dente. Drain, rinse briefly under cold water to prevent sticking, and set aside.
In a small bowl, whisk together the soy sauce, sesame oil, honey or brown sugar, rice vinegar, garlic, ginger, pepper, and cornstarch. Set aside.
Heat oil in a large pan over medium-high heat. Add the sliced beef and stir-fry for 2 to 3 minutes, until browned and just cooked through. Remove the beef from the pan and set aside.
In the same pan, add a little more oil if needed. Stir-fry the carrots and broccoli first for 2 to 3 minutes, then add the bell peppers and green onions. Cook until the vegetables are tender-crisp.
Add the cooked noodles and beef back into the pan. Pour the sauce over the top and toss until everything is evenly coated.
Stir-fry for another 1 to 2 minutes, until the sauce slightly thickens and coats the noodles. Garnish with extra green onions and serve hot.
Store leftover beef lo mein in an airtight container in the refrigerator for up to 4 days.