Heat a dry skillet over medium heat. Toast the guajillo and ancho chiles for 30 seconds per side until fragrant and slightly darkened. Transfer to a bowl and cover with 2 cups hot beef broth. Let soak for 15 minutes until softened.

Drain the soaked chiles, reserving the soaking liquid. Blend the chiles with the onion, garlic, oregano, cumin, vinegar, and 1 cup fresh beef broth until completely smooth, about 2 minutes. Strain through a fine-mesh sieve, pressing with a spoon to extract all the liquid.

Season the beef chunks with 1 teaspoon salt. In a large Dutch oven, sear the beef in batches over medium-high heat until deeply browned, about 3 minutes per side. Return all beef to the pot, pour in the chile puree, remaining 3 cups broth, and reserved soaking liquid. Bring to a boil, then reduce heat to low, cover, and simmer for 2.5 to 3 hours until the meat shreds easily with a fork.

Remove the beef with a slotted spoon and shred finely with two forks, discarding any large fat pieces. Skim excess fat from the broth. Simmer the broth uncovered over medium heat until reduced to about 2 cups of rich consomme, about 15 minutes. Taste and adjust salt. Reserve 1 cup consomme for dipping; the rest stays with the shredded meat.

Working with one wrapper at a time, place about 3 tablespoons shredded beef and 2 tablespoons Oaxaca cheese in the center. Fold the bottom corner up over the filling, tuck in the sides, and roll tightly toward the top corner. Brush the top edge with water to seal. Repeat with remaining wrappers, keeping assembled rolls covered with a damp towel.

Heat the vegetable oil in a heavy pot to 350 degrees F. Fry the egg rolls in batches of 3 to 4, turning occasionally, until deep golden brown and blistered, about 4 to 5 minutes. Maintain oil temperature between 340 and 360 degrees. Transfer to a wire rack set over a baking sheet to drain.

Serve the egg rolls immediately, with small bowls of the warm reserved consomme for dipping, plus cilantro and lime wedges on the side.