Season the beef with salt and pepper. Then toss with cornstarch coating well.
Heat a nonstick pan on medium-high heat. Once the pan is very hot, add the oil and beef, saute quickly to brown the beef then remove the beef with a slotted spoon, leaving the grease in the pan. Add the mushrooms and broccoli to the pan and saute until the mushrooms start to brown and soften, then add the garlic and ginger; saute for another 2 to 3 minutes.
Stir the broth, sesame oil, soy sauce, peanut butter and vinegar until well combined. Bring to a simmer and reduce to concentrate the flavor. Adjust the seasoning with salt if needed then remove from the heat.
Bring a large pot of water to a full boil. Cook noodles following the instructions on the package. Once cooked, drain and quickly rinse under cold water for a few seconds to remove any surface starch and stop the cooking process.
Return the pan to medium heat, toss the sauce with the noodles and beef. Garnish with scallions and red pepper flakes.