Preheat oven to 400 degrees. In a 12-inch skillet over medium heat, warm 1 tablespoon oil. Cook the onions, stirring occasionally, until lightly golden, about 5 minutes. Add the mushrooms; cook, stirring occasionally, until they're tender, 6 to 8 minutes.
Add spinach in two batches, letting the first batch wilt slightly before adding the next; cook until spinach is completely wilted and mixture is dry, 2 to 3 minutes total. Season with salt and pepper.
Brush one side of tortillas with remaining tablespoon oil. Place 4 tortillas, oiled sides down, on two baking sheets. Layer each tortilla with cheese, spinach mixture, and more cheese, dividing evenly. Top with the remaining 4 tortillas, oiled sides up, and lightly press to seal.
Bake until cheese has melted and tortillas are golden brown, turning them once, about 10 minutes. To serve, cut quesadillas into quarters.