Soak the noodles in warm water to soften, 5 to 10 minutes, then drain and snip into pieces with kitchen shears. Meanwhile, combine the soy sauce, 3 tablespoons sesame oil, the garlic, brown sugar and vinegar in a bowl. Put the beef in another bowl and toss with 2 tablespoons of the soy sauce mixture. Heat 2 teaspoons sesame oil in a large nonstick skillet over high heat. Add the onion and 1 teaspoon salt and stir-fry 2 minutes. Add the beef and stir-fry until just cooked through. Transfer to a bowl. Rinse and wipe out the skillet, return to the heat and add 2 teaspoons sesame oil. Add the mushrooms and carrots; stir-fry 3 minutes. Add the noodles and 2 tablespoons of the soy sauce mixture and stir-fry 1 minute; add 1/3 cup water and cook until the noodles are just tender, 3 minutes. Transfer to the bowl with the beef. Wipe out the skillet, return to the heat and add the remaining 2 teaspoons sesame oil. Add the spinach and the remaining soy sauce mixture and cook, stirring, 1 minute. Add to the beef and toss.