Put the sirloin chunks, short ribs, onion, tomatoes, chiles verdes, garlic, bay leaves, oregano, salt, and pepper in a large Dutch oven or deep lidded pot. Pour the 3 cups of water over the meat and bring to a boil over high heat. Cover, reduce the heat to medium low, and simmer for 2½ hours, stirring a few times in between, until the meat falls apart when pulled with a fork. Remove from the heat.
Remove and discard the bay leaves. Remove the meat and discard the bones, along with any tough connective tissue or gristle, then shred the meat into the remaining broth, along with the onion, tomato, garlic, and chile which will have cooked down to a paste.
Serve hot with warm pita bread, the Yogurt and Cucumber Salad with Herbs, Salsa Roja del Jacalito, and ripe avocado slices.