Gently reheat the shredded birria de res in a small pan over medium-low heat until warm. In a separate small saucepan, heat the birria consomé until it's simmering. Keep both warm while you prepare the sandwiches.

Lay the four slices of sourdough bread on a clean work surface. Generously butter one side of each slice. This buttered side will be the outside of your grilled cheese sandwich, creating a golden, crispy crust.

Flip two slices of bread so the unbuttered side is facing up. On one of these slices, layer half of the Monterey Jack cheese, followed by half of the Oaxaca cheese. This creates a good cheese base.

Evenly spread half of the warm shredded birria over the cheese layer. If desired, sprinkle half of the finely diced red onion and chopped fresh cilantro over the birria for added flavor and texture.

Cover the birria layer with the remaining half of the Monterey Jack and Oaxaca cheeses for the first sandwich. Place the second unbuttered bread slice on top, ensuring the buttered side is facing up. Repeat the assembly process for the second sandwich.

Heat a large non-stick skillet or griddle over medium-low heat. Maintaining a lower heat is crucial for ensuring the cheese melts completely without burning the bread.

Carefully place the assembled birria grilled cheese sandwiches into the preheated skillet. Cook for 5-7 minutes per side, or until the bread is a beautiful golden brown and crispy, and the cheese is completely melted and gooey. Adjust the heat as needed to prevent burning.

Once cooked, remove the birria grilled cheese sandwiches from the skillet. Let them rest for a minute to allow the cheese to set slightly, then slice them in half diagonally. Serve immediately with the warm birria consomé for dipping, along with lime wedges and a sprinkle of extra fresh cilantro.