Put chicken pieces in a large pan in one layer. Sprinkle with a little salt and cover with just enough water to all chicken pieces are submerged. Bring to a simmer and continue to cook, turning pieces over once, until almost done, about 20 minutes. Remove chicken pieces to a plate. Strain water that chicken was cooked in and return to pot.

Slit New Mexico chilies open and scrape out seeds and veins. Soak in hot water until soft, about 15 minutes. Drain, discarding soaking water.

Put chilies in a blender with 1 cup of water; blend until smooth. Set aside.

Blend garlic with 1/3 cup water and set aside.

Heat oil in a skillet and add jalapeño peppers. Saute a few seconds until crisp, but not burnt. Add pureed garlic and cook over high heat for 10 seconds until liquid has evaporated. Add New Mexico chili puree and cook, stirring and scraping bottom of pot until slightly reduced, about 5 minutes.

Add strained chicken cooking liquid and heat. Add chicken pieces to pan and cook over medium heat for about 25 minutes, turning pieces from time to time.

Remove bones and skin from chicken. Shred chicken.

Serve with warm corn tortillas (I warm them over my gas burner) and jalapeno pepper slices.