Heat 1 tablespoon of olive oil in a large skillet over medium heat. Add the sliced red onion and cook for 3-4 minutes until softened and translucent.

Add the chopped spinach, dried oregano, salt, and pepper to the skillet. Cook for 2-3 minutes until the spinach has completely wilted. Drain any excess liquid if necessary. Remove from heat and set aside to cool slightly.

Lay out the flour tortillas on a clean work surface. Sprinkle mozzarella cheese evenly over one half of each tortilla, leaving a half-inch border from the edge.

Divide the spinach and onion mixture evenly among the tortillas, spreading it over the mozzarella layer. Top with the crumbled feta cheese. Fold the empty half of each tortilla over the filling to create a half-moon shape, pressing down gently to seal.

Wipe out the skillet and heat the remaining 1 tablespoon of olive oil over medium heat. Once the oil shimmers, place one or two quesadillas in the skillet and cook for 3-4 minutes per side until golden brown and cheese is melted. Press down gently while cooking for even browning.

Remove the cooked quesadillas from the skillet and let them rest for 1-2 minutes before cutting into wedges. Repeat with the remaining quesadillas, adding more oil if needed. Serve immediately.