Boil the nix tamal in a saucepan of water until tender, about 30 minutes; then remove it from the boiling water and set aside.
Break the stems from the New Mexico chilies, shake the seeds from the pods, and wash the chilies. Add the chilies to the boiling water and boil until soft, about 15 minutes. When done, put the chilies in a blender with the 5 garlic cloves. Blend until smooth and then push through a sieve. Set the paste aside.
In a large saute pan over low heat, heat oil and then saute the onions and 10 crushed garlic cloves until soft, about 5 minutes. Add the sauteed onions and garlic to a large stockpot with the whole chickens, oregano, bay leaves, and chicken base. Cover with water, at least 3/4 of the way up the pot. Bring to a boil and then lower the heat and simmer until the chicken is cooked, about 20 minutes.
Remove the chickens from the broth and allow to cool. De-bone the chickens and then return to the broth. Add the cooked hominy and chili paste. Simmer for 30 minutes and then adjust seasoning with salt. Serve in large bowls garnished with sliced radish and sliced cabbage.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.