Heat the olive oil in a large skillet over medium heat. Add the finely diced onion and sauté for 3-4 minutes until soft. Stir in the minced garlic and cook for 1 minute until fragrant.
Add the cooked shredded chicken to the skillet. Sprinkle with chili powder, cumin, paprika, salt, and black pepper.
Pour in the salsa (or chicken broth). Stir constantly for 2-3 minutes until the chicken is evenly coated in the spices and heated through. Remove the filling to a bowl and wipe the skillet clean.
Lay the flour tortillas flat. Sprinkle a generous layer of shredded cheese over one half of each tortilla.
Spoon the seasoned chicken mixture evenly over the cheese. Top the chicken with another layer of cheese, then fold the empty half of the tortilla over to create a half-moon shape.
Return the skillet to medium heat and add a small amount of butter or oil.
Place 1-2 folded quesadillas into the skillet. Cook for 2-3 minutes until the bottom is deeply golden brown and crispy.
Carefully flip the quesadillas with a spatula and cook for another 2-3 minutes until the other side is golden and the cheese inside is completely melted.
Remove from the skillet, let rest for 1-2 minutes, then slice into triangles and serve warm with your favorite toppings.