Preheat the oven to 450°F and place a pizza stone inside if available.
Lightly flour a clean surface and stretch the pizza dough into a 12-inch circle.
Transfer the dough to a pizza peel or parchment-lined baking sheet and brush the crust edges with olive oil.
Sprinkle half of the mozzarella and Monterey Jack cheese evenly over the dough, leaving a ½-inch border.
Distribute the shredded birria meat over the cheese layer.
Top with the remaining mozzarella and Monterey Jack cheese.
Sprinkle half of the diced white onion and half of the chopped cilantro over the cheese.
Slide the pizza onto the hot stone or place the sheet in the oven and bake 12–18 minutes until the crust is golden and cheese is bubbly.
While baking, warm the birria consomé over low heat in a small saucepan until hot but not boiling.
Remove the pizza from the oven and let rest 1–2 minutes, then slice.
Garnish each slice with the remaining onion and cilantro, and serve with warm birria consomé for dipping.