In a large pot, bring water to a boil. Add dried noodles and cook according to the manufacturer's directions until tender with some chew (al dente). Alternatively, if using fresh noodles, skip the boiling step.
Drain noodles into a colander and rinse with cold water to stop the cooking process. Drain well and set aside.
In a small bowl, whisk together oyster sauce, soy sauce, sesame oil, vegetable broth, and cornstarch. Set aside.
Heat a large wok or nonstick skillet with high sides over medium-high heat. Once hot, add the vegetable oil. Once the oil begins to shimmer but not smoke, add in garlic and ginger. Stir-fry for 20 seconds, being careful not to burn the aromatics.
Add the cabbage and carrots, stir-fry until just tender, about 2 minutes.
Add the noodles and sauce. Toss and stir to combine, cook until sauce thickens and coats noodles, about 1 minute. If using fresh noodles, cook until softened and tender, which may require a few additional minutes.
Add bean sprouts and green onions, stir-fry until just tender, 1 minute. Serve while still hot.
Notes
Noodle Substitution: Yakisoba, ramen, lo mein, thin wonton noodles, thin or regular spaghetti.
Using Fresh Noodles: Skip the boiling step and add directly to the wok for par-cooked fresh noodles. Fresh noodles can be added directly to wok as directed. Raw fresh noodles need a few minutes of cooking, then drain well before adding. Toss in some oil if sticking together too much.
Soy Sauce Substitutions: Use coconut aminos or tamari.
Oyster Sauce Substitutions: Hoisin sauce or teriyaki sauce. Alternatively, 1/4 cup soy sauce and sugar (add 1 teaspoon sweetener at a time, increasing to taste). A small amount of fish sauce can enhance the missing seafood taste but use it sparingly, a 1/4 teaspoon to start.
For Saucier Noodles: Double the sauce ingredients.
Storing: Cool completely and store in an airtight container for up to 5 days. Reheat in the microwave, covered, in 30-second intervals until hot.