Warm shredded birria beef in a skillet or saucepan. Add a little consommé if needed to keep meat juicy.
Spread butter on one side of each bread slice.
Place bread butter-side down.
Layer mozzarella cheese, birria beef, and Oaxaca cheese on bread.
Add optional toppings if desired.
Top with second slice of bread, butter-side facing out.
Heat skillet over medium heat.
Cook sandwiches 3–4 minutes per side until bread is golden brown and cheese is melted.
Warm consommé in a saucepan.
Slice sandwiches and serve hot with consommé for dipping.