Prep the filling. Heat a large skillet over medium heat and add 1 teaspoon oil.
Sauté the onion and bell pepper with a pinch of salt for 3–4 minutes, until softened. Add the garlic and cook 30 seconds.
Season the chicken. Stir in the cooked chicken, chili powder, cumin, smoked paprika, and a few grinds of black pepper. Warm through for 2–3 minutes.
Add lime juice and cilantro, then taste and adjust salt.
Build the quesadillas. Transfer the filling to a bowl. Wipe the skillet if needed. Place a tortilla in the dry skillet over medium heat.
Sprinkle a thin, even layer of cheese on half the tortilla, add a layer of chicken mixture, then a little more cheese.
Fold and cook. Fold the tortilla over the filling to make a half-moon. Add a small pat of butter or a drizzle of oil around the edges for a crisp finish. Cook 2–3 minutes per side, pressing gently with a spatula, until golden and the cheese melts.
Repeat. Transfer to a cutting board and let rest 1 minute before slicing.
Repeat with remaining tortillas and filling, adding more oil or butter as needed.
Serve. Cut into wedges and serve hot with salsa, sour cream, guacamole, and lime wedges.