Prepare ingredients: thinly slice 1/2 small onion and half a bell pepper; dice the chicken breast; grate the cheese; dice the tomatoes and finely chop 1/2 onion for the pico.
Heat 1 tablespoon vegetable oil in a large skillet over medium heat. Add the sliced onion and bell pepper and cook until softened, about 4–6 minutes.
Add the diced chicken to the skillet with the cooked vegetables and cook until heated through and lightly browned, about 3–5 minutes; season with salt and pepper to taste.
Make the pico de gallo: in a mixing bowl combine the diced tomatoes, finely chopped 1/2 onion, minced jalapeño (if using), cilantro, lime juice, and salt and pepper to taste; stir and set aside.
Assemble the quesadilla: heat the skillet over medium heat and add the remaining 1 tablespoon oil or wipe the pan with oil. Place a 12-inch tortilla in the skillet, sprinkle half the shredded cheese over one side, top with the cooked chicken and vegetables, then sprinkle the remaining cheese on top.
Fold the tortilla in half and cook until the bottom is golden brown and the cheese begins to melt, about 2–3 minutes; flip and cook the other side until golden and the cheese is fully melted, another 2–3 minutes.
Remove the quesadilla from the skillet, let rest 1 minute, then slice into wedges. Serve topped with sour cream and the pico de gallo.