Soak the noodles: put the rice vermicelli in a large bowl, cover with lukewarm water, and soak for 8 to 10 minutes, until pliable but still firm in the center — they finish cooking in the broth later. Drain well. Noodles soaked to full softness now will turn mushy in the pan, so err on the firm side.
Sear the chicken and shrimp: heat a wok or a large, wide skillet over medium-high heat. Add 1 tablespoon of the oil, then the chicken in a single layer. Cook for 2 to 3 minutes until lightly golden and just cooked through; move it to a plate. Add the shrimp and cook 1 to 2 minutes, just until pink and curled. Add them to the plate with the chicken, leaving any flavored oil behind.
Build the aromatic base: add the remaining 1 tablespoon oil to the pan. Add the onion and cook for 2 minutes until softened at the edges. Add the garlic and stir for 30 seconds, just until fragrant — garlic goes bitter if it browns hard here.
Cook the vegetables in stages: add the carrot and green beans first and stir-fry for 2 minutes. Add the cabbage and toss for 1 more minute. The vegetables should be bright and crisp-tender; they soften a little more when the noodles and broth go in, so pull them back now rather than cooking them through.
Season the broth, then add the noodles: pour in the chicken broth, soy sauce, fish sauce, and black pepper. Bring it to a lively simmer and taste — it should be a little saltier than the finished dish. Add the drained noodles and toss with tongs, lifting and turning so every strand meets the broth. Cook for 3 to 4 minutes, tossing often, until the noodles have absorbed the liquid and turned supple and glossy. If the pan dries out before the noodles soften, splash in broth 1/4 cup at a time.
Bring it together and serve: return the chicken, shrimp, and any juices to the pan and toss for 1 minute to warm through. Taste and adjust: more fish sauce for salt and depth, more soy sauce for color. Pile the pancit onto a warm platter, scatter the scallions over the top, and serve right away with lemon wedges and extra black pepper.