Preheat oven to 425 °F (218 °C).
Line a rimmed baking sheet with foil.
Add shredded chicken and salsa to a large bowl; toss to coat. Add green onions, cilantro, diced green chiles, and cheese; gently stir to combine.
Lay tortillas out in a single layer on clean countertop or work surface. Spread ¼ of shredded chicken mixture over one half of a tortilla, leaving about a ½-inch border at the edge. Repeat with remaining shredded chicken mixture and tortillas. Fold tortillas over filling and press down to compact. Brush visible tortilla halves with 1 tablespoon oil. Arrange tortillas oiled side down on the prepared baking sheet in a single layer with round edge facing to the baking sheet center. Brush other side of tortillas with remaining 1 tablespoon oil.
Bake 10 minutes, or until quesadillas begin to brown. Carefully flip each quesadilla with a spatula, pressing down gently to compact. Bake an additional 5 minutes, or until golden brown. Transfer baking sheet from oven to cooling rack and let stand 5 minutes. With a pizza cutter or sharp knife, cut each quesadilla into 4 wedges and serve.
