Make the birria per recipe linked above. Reserve 1 cup of the stew into a serving bowl, and another 3 tbsp for topping the pizza. Remove and shred the beef.
Flatten and stretch dough into a 15-16" pie, sprinkling bread flour on the dough as needed to avoid sticking. If cooking in indoor oven, place dough on pizza peel prepared with a sheet of parchment paper. If using an outdoor oven, just place on a floured pizza peel.
Add 3 tbsp of the stew/consume onto the pizza as a sauce replacement. Starting from the outside and moving inward, spread a light even layer of cheese onto the dough. Evenly spread beef and white onion across pizza.
Choose ONE coothing method below.
(Option 1) Indoor oven
Preheat to 550F for 45 minutes with pizza steel or stone on bottom rack.Using the pizza peel, carefully move pizza to oven (parchment paper should make it slide easily - you can leave it on the paper or remove the paper after 1 minute of cooking), and cook for about 6 minutes, watching closely for crust color and cheese burn. Avoid rotating when baking in home ovens as temperature loss from opening the door often does more harm. If top needs more color, turn on broiler for 1-2 minutes towards the end.
(Option 2) Outdoor pizza oven
Preheat your Ooni (or other portable outdoor pizza oven) for 25 minutes. The oven should reach near 850F in this time. Reduce the flame to very low. Place pizza in oven and cook for ~3 minutes (+/- 30 seconds depending on color) rotating as needed. TIP: if using an Ooni or another model of outdoor pizza ovens, reduce the heat to low just as you place the pizza in; this creates really great crust color without burning the cheese too early. This may increase cook time slightly.
Let pizza cool for one minute on a wire rack or cutting board.
Top with cilantro. Serve with lime wedges and stew/consume to dip.
If using an outdoor pizza oven, such as Ooni Koda, the baking process will only take a couple minutes max. If using an indoor oven at 550F, it will take closer to 5-7 minutes.