Prepare your ingredients: grate the cheese on the large holes of a box grater. Cut the onions and mushrooms into thick slices, about 1/4-inch (7mm).
Cook the mushrooms: heat 1.5 tablespoons of oil/clarified butter in a large frying pan over high heat. Add the mushrooms and don't stir for about 2 minutes or until the mushrooms are browned at the bottom, then cook, stirring from time to time, until mushrooms are soft and browned (it will take about 10 minutes).
Cook the onions: heat 1.5 tablespoons of clarified butter/oil in a medium pot over medium heat, add the onions. Cook them for about 20 minutes, stirring from time to time, making sure they are not burning, until they are lightly caramelized - this means they are soft and lightly golden in color.
Make the filling: combine sauteed mushrooms, onions, and shredded cheese. Season with salt and pepper to taste (the cheese is already a bit salty).
Cook the quesadillas: Place some of the filling on a half of a tortilla. Fold the tortilla in half. Cook the quesadilla in a pan on both sides until golden. You can cook it in a dry pan or with a small amount of oil for a more crispy and delicious quesadilla. Heat about 2 teaspoons of oil in a frying pan over medium heat, when hot add the quesadillas, move with them a couple of times making sure they are not sticking to the pan. Cook them until golden on both sides and the cheese is melted.
Take the quesadillas off the pan and sprinkle lightly with salt.
Cut into two or four parts (depending on the size of quesadillas) and serve.
Enjoy!