In a large bowl, whisk together the salsa, sour cream, drained chopped green chiles, chili powder, cumin, garlic powder, onion powder, smoked paprika, dried oregano, and salt until evenly combined.
Add the shredded cooked chicken and both shredded cheeses to the seasoned salsa mixture; stir until the chicken and cheese are evenly coated. Taste and add cayenne pepper or hot sauce as desired.
If tortillas are stiff, stack 6 at a time, cover with a damp paper towel and microwave 60 seconds to make them pliable.
Spoon about 3 tablespoons of the filling onto the bottom third of each tortilla, then tightly roll the tortilla and place seam-side down; repeat with remaining tortillas and filling.
To fry: heat about 1 cup vegetable oil in a large nonstick skillet over medium-high heat until oil registers about 325°F (or is hot but not smoking). Fry 2–4 flautas seam-side down at a time, turning as needed, until golden brown on all sides, 3–4 minutes total.
Drain fried flautas briefly on paper towels, then transfer to a wire rack set in a baking pan and keep warm in a 200°F oven while you finish the rest.
Serve warm with sour cream, guacamole, salsa, pico de gallo, or shredded lettuce as desired.
