Heat 1 tablespoon of vegetable oil in a skillet over medium heat.
Add the thinly sliced half onion and diced half bell pepper to the pan and sauté until softened, about 4–6 minutes.
Dice or shred the cooked chicken breast and add to the pan; cook until heated through and lightly browned, about 3–4 minutes. Season with salt and pepper to taste.
Wipe the pan clean and return to medium heat. Place the 12-inch flour tortilla in the pan, sprinkle half the shredded cheese over one side, then top with the cooked chicken and vegetable mixture and the remaining cheese.
Fold the tortilla in half and cook 2–3 minutes per side, pressing lightly with a spatula, until the tortilla is golden brown and the cheese is fully melted.
Meanwhile, make the pico de gallo: combine the diced tomatoes, finely chopped onion, minced jalapeno (if using), chopped cilantro, lime juice, salt, and pepper in a bowl. Mix gently to combine.
Cut the quesadilla into wedges, serve topped with a dollop of sour cream and a side of pico de gallo.
