In a large skillet, heat the olive oil over medium heat.
Add the sliced onion and minced garlic, cooking for about 5 minutes until the onion is soft and fragrant, stirring occasionally.
Add the chopped mushrooms to the skillet and cook for another 5 minutes, stirring until they are nearly tender.
Add the fresh spinach in batches, stirring continuously until it has just wilted.
Remove the skillet from the heat.
Stir the mozzarella cheese, Parmesan cheese, and basil into the spinach and mushroom mixture until well combined.
Spoon one-quarter of the filling onto one half of each tortilla.
Fold the other half over the filling, pressing down gently to create a half-moon shape. Lightly coat both sides of each quesadilla with cooking spray.
Wipe the skillet clean and return it to medium heat.
Cook two quesadillas at a time for 2 to 3 minutes per side, or until the tortillas are golden brown and the cheese is fully melted.
Transfer to a baking sheet and keep warm in a 300°F / 150°C oven while you cook the remaining two.
Slice in half and serve immediately.