Heat the birria consomé in a shallow bowl. Dip one side of each tortilla briefly into the consomé to lightly coat it with the flavorful oil and broth.

On the consomé-dipped side of each tortilla, add a layer of shredded Oaxaca and Monterey Jack cheese on half of the tortilla. Top with shredded birria beef, diced onion, and cilantro.

Fold the tortilla in half to create a half-moon shape, pressing gently to seal. Repeat with remaining tortillas.

Heat a large skillet or griddle over medium-high heat and add a thin layer of oil. Place 2-3 quesadillas in the pan and cook for 3-4 minutes per side until golden brown and crispy, and the cheese is melted.

Transfer cooked quesadillas to a plate and keep warm. Repeat with remaining quesadillas, adding more oil as needed.

Heat the remaining birria consomé in a small pot until hot and steaming.

Cut each quesadilla in half and serve immediately with warm consomé for dipping, lime wedges, extra onion, and cilantro on the side.