In a medium saucepan set over medium heat, combine the meat with 2 quarts salted water, about 1/3 of the onions, and half of the garlic and simmer until the meat is very tender, about 1 1/2 hours. Strain, reserving the broth for another use. When the meat is cool enough to handle, shred it into coarse strands with your fingers or 2 forks--don't worry that there are bits of onion and garlic mixed with the meat.
Wash and dry the saucepan, set it over medium heat and add 1 tablespoon oil. When the oil is hot, add half of the remaining onions and cook until golden, about 6 minutes, then stir in the remaining garlic and cook for another minute. Add the tomatoes and chiles and cook until most of the juice has evaporated, about 3 minutes. Stir in the shredded meat and simmer for a few more minutes, then taste and season with about 1/2 teaspoon salt. Remove from the heat.