Preheat the oven to 450°F. Place a 10-inch oven-proof skillet near the stove.
In a large bowl, combine the Queso Oaxaca, Pepper Jack, and Mozzarella cheeses and set the bowl aside.
Heat the 10-inch skillet over medium heat. Add the chorizo and cook, breaking it into pieces and stirring occasionally, until crisp and fully cooked, about 8–10 minutes.
Using a slotted spoon, transfer the cooked chorizo to a separate bowl, leaving the drippings in the skillet.
Add the chopped onions to the skillet with the drippings and cook, stirring, until soft and translucent, about 3–4 minutes. Add the minced garlic and cook, stirring, for 30 seconds.
Scrape the onion-and-garlic mixture from the skillet into the bowl with the chorizo and stir to combine. Scoop out 1/4 cup of this chorizo-onion mixture and set it aside for topping.
Add the 4 oz can of diced green chilies to the remaining chorizo-onion mixture and stir to combine.
Sprinkle half of the mixed cheeses evenly over the bottom of the same 10-inch skillet.
Spread the chorizo-and-chile mixture evenly over the cheese layer in the skillet.
Top the chorizo mixture with the remaining half of the mixed cheeses, spreading evenly.
Bake in the preheated 450°F 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 hot.