Whisk together all the sauce ingredients in a large mixing bowl. Divide in half, reserving one half for cooking. Add the beef to the remaining half and toss to coat. Refrigerate to marinate meat at least 20 minutes, and up to overnight. Stain and discard marinade.

While the meat marinates, cook the noodles according to package directions. Strain and set aside. Heat the vegetable oil over medium-high heat until shimmering. Add meat and stir-fry until it begins to brown on the outside. Add broccoli and cook just until broccoli turns bright green. Add remaining half of sauce and cook just until it begins to simmer. Remove from heat and add noodles, tossing to coat in sauce. Serve immediately, with fresh grated black pepper over the top.