Mix both mozzarella and cheddar cheese with chopped cilantro and green onions in a medium bowl.
Top one tortilla with a quarter of the cheese mixture and a quarter of the pulled pork, then cover with another tortilla. Make sure you layer the cheese and pork. You should start and finish with the cheese to ensure tortillas and cheese are in contact on both sides. Otherwise the quesadillas will come apart.
Heat the oil in a frying pan, then add a quesadilla and cook over medium heat until golden for approximately 3-5 minutes (depending on the heat source), flip it and cook for 3 minutes longer. Your quesadillas are ready when the cheese is melted all the way through and the tortillas are golden and crispy. Repeat with the remaining quesadillas. You might need to add more oil if needed.
You can skip the oil if using a non-stick frying pan.