Remove stems and seeds from dried chiles, then soak them in hot water for 10 minutes until softened and pliable.
Blend softened chiles, garlic, onion, cumin, oregano, paprika, salt, pepper, vinegar, and 1 cup broth until smooth and deep red.
Place beef in a Dutch oven, pour sauce over it, add remaining broth, and simmer covered on low for about 3 hours until fork-tender and shreddable.
Shred beef using two forks and return it to the cooking liquid to stay juicy; reserve 1 cup of consommé for dipping.
Flatten each biscuit dough round into a 4-inch circle, add 2 tablespoons shredded beef and 1–2 tablespoons cheese, then pinch tightly to seal into a smooth ball.
Heat oil to 350°F; fry bombs in batches for 2–3 minutes, turning until evenly golden brown and crisp.
Remove with tongs and drain on a baking sheet lined with paper towels until exterior is crunchy and deeply golden.
Serve hot with chopped cilantro and warm consommé for dipping.
