Add the beans, 10 cups of water, ham shanks, the prepared sofrito, and the chicken base to a large pot, stir well to combine.
Set the pot over high heat and bring the water to a boil. Then, reduce the heat to medium-low, cover, and simmer for 1 hour. Stir occasionally and keep it at a simmer, raise or lower the heat as needed.
Uncover the beans, remove the smoked ham, and set them on a plate or cutting board to cool.
Cover the beans and continue cooking while you chop the ham.
When the smoked pork is cool enough to handle, remove the meat from the bone. Discard the bones and any fatty pieces. Chop the meat into bite sized pieces and return it to the pot.
Continue cooking the beans (covered) for another 15-20 minutes. Try a couple of beans to determine how close to done they are. If the beans are still quite firm, cover the pot and cook for 30 minutes. Then, try them again.
When the beans are soft, uncover the pot to allow the liquid to thicken. This process can take from 35-55 minutes, depending on how much liquid is in the pot and how thick you want them.
Keep the beans at a simmer. Raise or lower the heat as needed. Stir occasionally at first then as they thicken stir more often to keep from burning on the bottom.
Taste the beans and add salt if needed as a reference we added ¼ teaspoon to ours. Serve with white rice or crusty bread, if desired.