Step 1Heat 1 tablespoon oil in a large skillet. Add the onion and cook until it softens and begins to color, 3 to 5 minutes.
Step 2While the onion is cooking, rinse the greens in a colander but don't dry them. When the onion is ready, add the greens along with any water that clings to the leaves to the skillet. Season with salt and red pepper, sprinkle with garlic and cook, stirring frequently, until the greens are quite tender, about 15 minutes.
Step 3Remove from the heat and stir in the feta.
Step 4Heat 2 teaspoons oil on a griddle over medium heat; use a spatula to distribute it so the surface is evenly covered. If you don't have a large griddle, you can use a skillet, but you'll have to cook the quesadillas in more than one batch. Add the tortillas and cook on one side until they have softened, 2 to 3 minutes. Flip them to the other side and spoon 2 to 3 tablespoons of the greens in the center. Sprinkle over some of the diced mozzarella and use a spatula to fold the tortilla in half around the filling.
Step 5Cook until the tortilla begins to brown on one side, 2 to 3 minutes, and then flip onto the other side and cook until the tortilla browns on that side and the mozzarella is melted, about 2 to 3 minutes more. If some of the melting cheese oozes onto the griddle, that's even better. Repeat if necessary to use up all of the tortillas and filling.
Step 6Serve immediately, or keep warm in a 250-degree oven until all of the quesadillas have been made.