Bring a large pot of heavily salted water to a boil. Add noodles and cook according to package directions. Add broccoli for the last 5 minutes of boiling and let cook until broccoli is fork tender. Drain, rinse with cold water, and drain again.
In a small bowl, whisk together brown sugar, lime juice, and soy sauce. In a large skillet, heat oil over medium-high heat. Add scallions and garlic and cook, stirring constantly, until fragrant. Add soy sauce mixture and noodles and broccoli to the skillet. Toss constantly until noodles and broccoli are coated with the sauce.
Serve noodles with lime wedges and garnish with cilantro and peanuts.
Note from poster- I add about 12 oz of tofu to this proportion, and I put more sauce than stated here.