Coarsely chop the leftover cooked chicken or turkey, or place it in a food processor.
Pulse the chicken in the food processor a few times until shredded; alternatively chop and shred by hand.
Add the drained diced green chiles and the cream cheese to the shredded poultry and pulse until the mixture is well combined and spreadable.
Heat a nonstick frying pan or skillet over medium-low and brush or mist lightly with olive oil.
Spread about half of one tortilla with the chicken and green chile mixture, sprinkle with a little grated mozzarella, then fold the tortilla over to form a half-moon.
Place the folded quesadilla in the pan and cook about 2 minutes per side, or until the cheese is melted and the tortilla is golden brown.
Repeat with remaining tortillas and filling, adding more oil to the pan as needed.
Serve hot with sour cream and salsa on the side, if desired.