Start by heating one tablespoon of oil in a large pot over medium heat. Sear the beef shanks on both sides until they develop a nice brown color, about 3-4 minutes per side. Remove the beef and set aside.
Pour the water into the same pot and bring it to a boil. Return the beef shanks to the pot and skim off any gray foam that rises to the surface. Once the broth looks clear, add the quartered onions, whole garlic cloves, peppercorns, and salt. Lower the heat to a gentle simmer, cover the pot, and let it cook for about 1½ to 2 hours, or until the beef is fork-tender.
While the beef is cooking, prepare your berenjana sauce. Place the whole eggplants directly over an open flame on your stove, turning occasionally until the skin is completely charred. Run them under cold water and peel off the blackened skin. Mash the soft eggplant flesh in a bowl, then mix in the minced garlic, vinegar, oil, and salt to taste. Set this sauce aside.
In a separate wide pan, heat another tablespoon of oil. Add the saba bananas and cook until they're lightly golden brown on all sides. Remove them and set aside. In the same pan, cook the chorizo de bilbao for 1-2 minutes until it releases its red oil. Remove and set aside. Add the chicken pieces to the pan and cook until they're lightly browned on all sides.
Once the beef is tender, add the browned chicken, ham slices, and chorizo to the pot. Cook for about 15-20 minutes until the chicken is fully cooked through. Add the garbanzo beans and browned saba bananas, letting them cook for 3-5 minutes to heat through.
Next, add the green beans and cabbage quarters to the pot. Cook until they're tender but still have a slight crunch. Finally, add the pechay leaves and cook for just one more minute. Taste the broth and add more salt if needed.
To serve your cocido the traditional way, first ladle the hot, clear broth into bowls as your first course. Then arrange all the vegetables and garbanzo beans on a large platter for your second course. Finally, present all the meats on another platter for the third course. Serve everything with steaming white rice and the berenjena sauce on the side.
Your cocido is best enjoyed fresh, but if you have leftovers, store everything together in the broth to keep the meat and vegetables from drying out. It will keep well in the refrigerator for up to three days.