Begin by placing chicken breasts in a large pot with quartered onions, salt, peppercorns, and bay leaves. Add cold water until it covers the chicken by one inch. Bring to a boil, then reduce heat to low and simmer covered for 10-15 minutes until chicken is cooked through. Remove chicken, let it cool, then shred it into pieces. Strain the liquid and save 2 cups of this broth.
While the chicken cooks, soak your cellophane noodles in warm water until they soften. Once soft, drain and set aside. This is also a good time to prepare your vegetables.
Heat vegetable oil in a large wok or wide pan over medium heat. Add chopped onions and minced garlic, cooking until the onions become clear and soft. Add ground pork and cook until it turns light brown, breaking up any lumps as it cooks.
Add your carrots and green beans to the pan. Cook for one minute. Add the sliced cabbage and cook for another minute. The vegetables should still be crisp at this point.
Pour in the oyster sauce and sesame oil, stirring everything together. Now add both the softened cellophane noodles and fresh miki noodles to the pan. Pour in one cup of your reserved chicken broth, add the shredded chicken, and gently toss everything together using tongs.
Add more broth as needed if the noodles look dry. Keep gently tossing until the noodles are cooked but still firm and most of the liquid has been absorbed. The vegetables should be tender-crisp. Taste and add salt and pepper if needed.
Serve hot, garnished with toasted garlic and green onions if you like. Place calamansi halves on the side for squeezing over individual portions.