Preheat oven to 450°F (232°C).
In a large bowl, mix the Queso Oaxaca, Pepper Jack, and Mozzarella cheeses together; set aside.
Place an oven-proof 10‑inch skillet over medium heat. Add the chorizo and cook, stirring occasionally, until crisp and fully cooked, about 8–10 minutes.
Using a slotted spoon, transfer the cooked chorizo to a large bowl, leaving the pan drippings in the skillet.
Add the chopped onions to the same skillet and cook until soft and translucent, about 3–4 minutes. Add the minced garlic and cook an additional 30 seconds.
Scrape the onion-and-garlic mixture into the bowl with the chorizo. Scoop out and reserve 1/4 cup of this chorizo-onion mixture for garnish.
Add the 4 oz can of diced green chilies to the remaining chorizo-onion mixture in the bowl and stir to combine.
Evenly sprinkle half of the mixed cheeses into the bottom of the skillet.
Spread the chorizo-and-chiles mixture over the cheese in the skillet.
Evenly spread the remaining cheese on top of the chorizo layer.
Bake in the preheated oven until the cheese is melted and bubbly, about 20 minutes.
Remove from the oven, sprinkle the reserved 1/4 cup chorizo-onion mixture on top, and garnish with chopped cilantro and chopped green onions. Serve warm.