Heat 1 tablespoon of olive oil in a large non-stick skillet over medium heat. Add the finely diced red onion and sauté for 3-4 minutes until softened and translucent. Add the minced garlic and cook for another minute until fragrant.

Add the fresh baby spinach to the skillet. Cook, stirring occasionally, until the spinach has completely wilted down, about 3-5 minutes. Season with dried oregano, salt, and black pepper. Remove the skillet from heat and let the spinach mixture cool slightly for 5 minutes.

In a medium mixing bowl, combine the cooled spinach mixture with the crumbled feta cheese and shredded mozzarella cheese. Mix everything thoroughly until well combined.

Place one large flour tortilla on a clean work surface. Spread about 1/4 of the spinach and cheese filling evenly over the entire surface of the tortilla, leaving about a 1/2-inch border around the edges. Top with another flour tortilla, gently pressing down.

Heat the remaining 1 tablespoon of olive oil in a large non-stick skillet or on a griddle over medium heat. Once the oil is hot, carefully place one assembled quesadilla into the skillet.

Cook for 3-4 minutes per side, or until the tortilla is golden brown and crispy, and the cheese is completely melted and bubbly. You may need to press down gently with a spatula to ensure even browning and melting. Transfer the cooked quesadilla to a cutting board.

Repeat the cooking process with the remaining quesadillas, adding a little more olive oil to the pan if needed between batches.

Once all quesadillas are cooked, slice each one into 4 or 6 wedges. Serve immediately with your favorite Mediterranean-inspired dipping sauce, such as tzatziki, plain Greek yogurt, or a lemon-dill sauce.