Bring a large pot of water to a boil.
Cut steak in half lengthwise, with grain then cut each half against grain into thin ¼-inch pieces; season with 1 teaspoon pepper and ½ teaspoon kosher salt.
Heat 1 tablespoon oil in 12-inch nonstick skillet over high heat until shimmering.
Add beef and scallion white and light green parts to skillet; use tongs to arrange in a single layer. Cook, without stirring, until beef is browned on bottom, about 2 minutes.
Stir and continue to cook until beef is no longer pink, 2–3 minutes longer, stirring occasionally.
Meanwhile, drop noodles in boiling water and cook according to package directions. Drain and set aside.
For sauce, whisk together ½ cup oyster sauce, ½ cup broth, 3 tablespoons tamari, ¼ cup ginger, 2 tablespoons garlic, and 2 tablespoons honey. Add 2 ½ tablespoons cornstarch and whisk until smooth.
Add sauce mixture and broccoli to skillet and toss to combine. Reduce heat to medium, cover, and cook until broccoli is crisp-tender, 3–5 minutes, uncovering and stirring beef and broccoli occasionally.
Off heat, add cooked noodles and scallion greens and toss to combine.