Meanwhile, heat oil on medium-high in an extra-deep, 12-inch skillet. Peel and coarsely chop onion, adding to skillet as you chop. Stir occasionally. While onion cooks, slice steak about 1/8-inch thick and add to skillet. Add mushrooms, broccoli, garlic and ginger. Cook for 4 minutes, stirring frequently.
Add all remaining ingredients. Stir well. Remove skillet from heat. When noodles are done, drain well and add to skillet. Stir until noodles are combined well with vegetables and coated with sauce. Serve at once.