Heat broiler. Heat the oil in a large skillet over medium-high heat.
Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper and cook until golden brown and cooked through, about 2 minutes. Cut into strips.
Place the tortillas on a broilerproof rimmed baking sheet and broil until crisp, about 1 minute.
Top the tortillas with the chicken, chorizo, and Cheddar. Broil until the cheese melts. Serve with the sour cream and cilantro.