Heat olive oil in a large skillet over medium heat. Add diced onion and bell pepper, sautéing for 5-7 minutes until softened.
Incorporate shredded chicken, chili powder, cumin, garlic powder, and salt into the skillet. Stir for 3 minutes until chicken is well-coated and heated through, then transfer mixture to a bowl.
Wipe the skillet clean and return to medium heat, melting 1/2 tablespoon of butter.
Place one flour tortilla in the skillet. Sprinkle 1/4 cup of cheese over one half of the tortilla.
Distribute 1/4 of the chicken and vegetable mixture over the cheese, then top with an additional 1/4 cup of cheese and a sprinkle of cilantro.
Fold the tortilla in half and cook for 2 to 3 minutes per side until the tortilla is golden brown and the internal cheese is fully melted.
Repeat the assembly and cooking process for the remaining tortillas.
Allow quesadillas to rest for 2 minutes before slicing into wedges. Serve warm with salsa and sour cream.