Preheat the oven to 325 degrees F.

In a small skillet, cook the chorizo over medium heat until browned, about 5 minutes, breaking it into pieces with a wooden spoon. Remove from heat and drain on a paper towel-lined plate. In a deep mixing bowl, whisk together the eggs and milk until well beaten. Pour the egg mixture into the unbaked pie skin. Evenly spoon the chorizo and potatoes into the egg-filled pie skin. Arrange the green chile strips in a sunburst pattern on top of the quiche filling. Sprinkle queso fresco over top. Bake for 40 minutes. Slice into 4 wedges and serve immediately, with a warm tortilla, if desired.