Pour oil into a 10- to 12-inch frying pan over medium heat. When hot, add Mexican chorizo, potatoes, mushrooms, onion, and jalapeño chili; stir often until meat is well browned and potatoes are tender when pierced, about 10 minutes (if potatoes aren't tender, add 1/4 cup water, cover pan, and cook until tender, 8 to 12 minutes longer). Pour in eggs, beaten to blend, and stir until eggs are firm, 1 to 2 minutes.
Remove pan from heat and stir in cheese. Serve with warm tortillas and lettuce, sour cream, salsa, and chopped onions to add to taste.