Prepare the pork and shrimp: Preheat a wok over high heat until hot. Add the peanut oil and swirl to coat. Add the pork and shrimp and cook, stirring, for 1 minute. Add the soy sauce and oyster sauce; season with salt and pepper. Cook, stirring, for another minute or so. Transfer the pork and shrimp to a plate and set aside to cool. Wipe out the wok.
Meanwhile, make the omelets: Beat the eggs in a large bowl. Add the cabbage, carrot, scallions, peas, bean sprouts, ginger, and the reserved pork and shrimp and stir well to combine. Do not season; salting now will make these omelets watery.
Preheat the wok over medium heat. Add 2 tablespoons of the oil and one-fourth of the omelet mixture and cook, until the egg is set, about 1 minute. If needed, add a bit more of the beaten egg, pushing in the edges to make the omelet round. Flip and cook until the omelet starts to tighten and brown at the edges, 1 minute. Transfer to a plate. Repeat to make 3 more omelets, adding more oil to the wok as needed. Season the egg foo yung with salt and serve with one or both of the sauces.
Egg Foo Yung Sauce:
Whisk together all of the ingredients and heat in a small saucepan until simmering. Remove from the heat, cover and keep warm or let cool, then refrigerate up to 3 days. Rewarm before serving.
Bring a medium saucepan of water to a simmer. Meanwhile, combine all of the ingredients except the sesame oil in a medium stainless steel bowl. Place the bowl over the simmering water in the saucepan (do not let the bowl touch the water). Whisk occasionally until the yolks set, 5 to 7 minutes. Whisk in the sesame oil. Taste and add more sugar, if desired. Cover and keep warm or let cool, then refrigerate up to 3 days. Rewarm gently before serving.