Heat 1 tortilla in a large nonstick skillet over medium heat until warm, about 1 minute. Sprinkle with one-quarter each of the cheese, chicken, avocado, scallions and a pinch of salt and pepper.
Cover and cook until the cheese is melted, about 2 minutes. Transfer to a cutting board, fold tortilla in half, and place on a baking sheet in the over to keep warm. Repeat to use remaining ingredients.
Cut each quesadilla into 3 triangles and serve with salsa.