Rinse beef and pat dry; season well with salt. In a large pot, heat oil over high heat. Add beef and brown on all sides, 6 to 8 min. Transfer beef to a plate. Reduce heat to medium; add onion and garlic. Cook until onion is softened, about 3 min. Return beef to pot; add broth, tomatillos, juice and cumin. Reduce heat to low, cover and simmer 1 hr. Transfer beef to a cutting board and, when cool enough to handle, shred meat (discard bones) and return to pot. Add chile, carrot, celery and chayote; simmer until tender, 20 to 25 min more. Skim off fat and stir in cilantro.
Ladle soup into bowls and sprinkle with cheese. Serve with tortilla chips (if using).
Beef short ribs are an fine sub, as is ossobuco (but that can be pricey and doesn't quite have the same flavor). In a pinch you can also use chuck roast. Cut it into pieces before searing it.