Tear or shred the cooked chicken into bite-sized pieces. In a medium bowl, combine shredded cheddar and Monterey Jack cheeses, shredded chicken, chopped cilantro, ground cumin, smoked paprika, kosher salt, and black pepper. Toss until evenly mixed.

Sauté vegetables

Heat 1 tablespoon oil in a medium nonstick or cast-iron skillet over medium heat. Add the thinly sliced red onion and bell pepper and sauté, stirring occasionally, until softened and slightly caramelized, about 4–5 minutes. Remove vegetables to a plate.

Assemble quesadilla

Wipe the skillet if necessary and return to medium-low heat. Lay one flour tortilla flat on a cutting board. Evenly spread about 1/4 of the cheese-and-chicken mixture over half of the tortilla, top with a portion of the sautéed vegetables, then fold the tortilla in half to enclose the filling. Repeat with remaining tortillas and filling.

Cook quesadilla

Melt 1/2 tablespoon butter (or brush 1/2 tablespoon oil) in the preheated skillet over medium-low. Carefully transfer one folded quesadilla to the skillet. Cook until the bottom is golden brown and the cheese begins to melt, about 2–3 minutes. Using a spatula, flip and cook the other side until golden and cheese is fully melted, another 1.5–2 minutes. If needed, press gently with the spatula to help the tortilla crisp and the cheese bind. Repeat with remaining quesadillas, adding more butter or oil as needed.

Finish and slice

Remove cooked quesadillas to a cutting board and let rest 1 minute. Slice each into 3–4 wedges with a sharp knife or pizza cutter. Serve hot with sour cream, salsa, avocado or guacamole, and lime wedges.