Heat oil in a 10-inch, oven-safe skillet. Cook chorizo for 7–8 minutes at medium heat, until the fat renders and the sausage is bubbly. Turn off heat, add the green chiles to the pan, and stir to combine with chorizo.

Drain fat from the skillet. After draining, place the mixture on a paper towel–lined plate to remove as much fat as possible. Reserve 3 tablespoons of mixture for garnish and return mixture to skillet; spread in an even layer.

Preheat oven broiler with a rack 4 inches from the heating element. Top the chorizo with the grated cheese. Broil until bubbly and browned in spots, 2–3 minutes. Garnish with reserved chorizo. Serve immediately with your favorite tortillas or tortilla chips.

Make ahead: Cook chorizo and chiles up to a day ahead; store in an airtight container in the refrigerator. When ready to serve, warm the meat mixture in a skillet, top it with cheese, and broil it when your guests arrive.