Preheat oven to 350 degrees F (175 degrees C). Sprinkle 1/2 the cheese on one side of each tortilla. Place tortillas cheese side up on baking sheets, and bake 5 minutes, or until cheese is melted.
Melt the butter in a skillet over medium heat. Stir in garlic and mushrooms, and cook about 5 minutes. Mix in spinach, and continue cooking until spinach is slightly wilted. Place an equal amount of the mixture on the cheese side of each tortilla.Place tortilla on top of tortilla. sprinkle with remaining cheese.
Heat oil in a separate skillet over medium heat. Place quesadillas in the skillet one at a time, and cook 3 minutes on each side, until golden brown. Cut each quesadilla into 4 wedges to serve.