Heat the oil in a large skillet over a medium heat. Add the onions and mushrooms and cook until the mushroom water is evaporated and they begin to brown, 5 to 7 minutes. Add the garlic and cook for 1 minute more. Add chicken, cumin, chili powder and oregano and stir until all spices are incorporated. Add spinach, salt and pepper and cook until spinach is wilted, about 2 minutes.
Lay 1 tortilla on a flat work surface and sprinkle with 1/4 cup shredded cheese. Spoon 1/2 chicken and vegetable mixture on top of cheese, then top with an additional 1/4 cup cheese. Top with another flour tortilla. Heat a large nonstick skillet with cooking spray over medium heat. Carefully place 1 quesadilla in pan and cook 3 minutes. Using a large spatula, gently flip quesadilla and cook an additional 3 minutes until lightly browned and cheese is melted. Repeat with second quesadilla. Slice each quesadilla into quarters. Place 2 quarters on a plate with 1 tablespoon sour cream and 2 tablespoons salsa.