The Shredded beef filling: 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 of the 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 and set aside.
Forming, filling, and griddle-baking the gorditas: Heat a well-seasoned or nonstick griddle or heavy skillet over medium heat. Knead the masa (fresh or reconstituted) to make it pliable, adding a little water if necessary to achieve a soft-cookie-dough consistency. Knead in the flour, baking powder, and 3/4 teaspoon salt. Divide the dough into 10 portions and roll into balls; cover with plastic to keep from drying out. Line a tortilla press with 2 pieces of plastic cut to fit the plates (to be on the safe side, cut them from a food storage bag; the thicker plastic usually works better for beginners). Gently press out a ball of dough between the sheets of plastic to about 4 inches in diameter (it'll be about 1/4 inch thick). You've now made a gordita, which is what you call a fat tortilla. Peel off the top sheet of plastic, flip the gordita, uncovered side down, onto the fingers of 1 hand, and gently peel off the second piece of plastic. In one flowing movement, roll the gordita off your hand and onto the heated griddle or skillet. Bake for about 1 1/2 minutes, then flip and bake for another 1 1/2 minutes on the other side. The gordita will be lightly browned and crusty on the top and bottom, but still a little uncooked on the sides. Remove to a plate. Continue pressing and griddle-baking the remaining gorditas in the same manner.
Finishing the gorditas: When you're ready to serve, warm the shredded beef. Rinse the remaining onions in a small strainer under cold water and shake to remove the excess moisture. Have the cheese and cilantro at the ready.
In a deep heavy medium skillet or saucepan, heat 1/2-inch of oil over medium to medium-high until the oil is hot enough to make the edge of a gordita sizzle sharply, about 350 degrees F on a deep-fry thermometer. One by one, fry the gorditas, turning them after they've been in the oil for about 15 seconds, until they're nicely crisp but not hard, about 45 seconds total. When they're ready, most will have puffed up a little, like pita bread. Drain on paper towels.
Once they all are fried, use a small knife to cut a slit in the thin edge of each one about halfway around its circumference, opening a pocket. As you cut them, fill each gordita with about 1/4-cup shredded meat and a sprinkling of the onions, grated cheese, and cilantro.
Line up the filled gorditas on a serving platter and pass them around (with plenty of napkins) for your guest to enjoy.