Rather than tortillas rolled around a filling, then topped with sauce, New Mexico enchiladas are stacked, with layers of tortilla, chile sauce, cheese, and fillings. They can be as simple as cheese and chile sauce or they can incorporate more substantial fillings.
Have ready:
Heat in a medium skillet over medium heat (dip the edge of a tortilla in the oil and it should sizzle):
Fry the tortillas briefly, flipping once, until they have puffed slightly on both sides, about 30 seconds total. Transfer to paper towels to drain. Have ready:
Adjust a rack so it is 6 to 8 inches below the heating element. Preheat the broiler. Line a large baking sheet with foil.
To make the enchilada stacks, set 4 tortillas on the baking sheet. Top each with ¼ cup filling, if using, then 2 heaping tablespoons of the sauce, followed by ¼ cup of the cheese. Top each stack with a second tortilla, ¼ cup optional filling, 2 heaping tablespoons sauce, and ¼ cup cheese. Top each stack with a third tortilla, then divide the remaining sauce and cheese among them.
Broil until the cheese is melted, 3 to 5 minutes.