Spray a large heavy skillet with nonstick cooking spray and set on moderately high heat until hot but not smoking. Add the ground beef and brown for about 10 minutes. Drain off any excess grease.
Add the chorizo by crumbling it into the cooked ground beef. Cover the skillet and cook the chorizo thoroughly for another 5 to 10 minutes. Using a potato masher, combine the beef and chorizo together.
Transfer to a large pot and add the frijoles and liquid and cook for 5 minutes. Add the red chile sauce, water, and oregano and cook until it starts to boil, stirring occasionally. Reduce heat to low and simmer, uncovered, for about 10 minutes. Add more liquid from beans if mixture is too thick.
Add salt to taste. Ladle into bowls and sprinkle with a handful of cheese.
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