S-A here's the thread with my Hatch Green Chile Sauce. I think you could add about 1/2-1 cup of shredded jack cheese slowly as the sauce begins to thicken. You may need a little more stock to compensate. I'd also add about 1/2 cup of heavy cream. Add in cilantro and lime juice and hit with a blender. I would think that would get you pretty close if not better than Chuy's
http://www.texags.com/main/forum.reply.asp?topic_id=1470949&forum_id=12Chile Verde ( Basic Green Chile Sauce) makes 2 cups +
2 Tbs vegetable oil or olive oil
1/4 to 1/2 small onion, diced
1 large garlic clove, minced
2 Tbs flour
1/4 tsp ground cumin
1/4 tsp black pepper
1 1/2 cups chicken stock
1 cup roasted, peeled green chiles (10-12 fresh New Mexico, Anaheim or California or poblano chiles), or (if you don't have fresh chiles) 8 oz canned green chiles.
1/4 tsp oregano
1/2 tsp salt
(optional: 2 tsp or more chopped, canned or fresh jalapenos if you want
it hotter)
Heat the oil in a 1-2 qt saucepan over medium heat. Add onion and garlic, cover and cook over low heat for about 5 minutes to wilt the onions. Don't let them brown. Raise heat to medium, stir in flour, cumin and black pepper. Cook, stirring about 2 min. When the onion/flour mix just begins to color, remove from heat and add broth, stirring constantly to avoid lumps.
Add rest of ingredients. Return pan to heat, bring to a boil, cover and simmer over low heat another 30 min. The finished sauce should not be too thick; dilute with more broth or water if needed.
You can puree it if it's not smooth enough.