Tortillas: Mix masa harina with warm water. Using your hands squish water and flour together until a cohesive mass is formed. Dough should be smooth and pliable. Cover the bowl with a barely damp kitchen towel. Let sit 5-10 minutes. Add salt; knead gently in bowl for another minute. Divide dough into 12 equal portions. Use your palms to form each portion into a ball. Cover with the damp towel.
Filling: In a frying pan over medium heat, warm the oil. Add onion and sauté until golden brown about 5 minutes. Stir in garlic and oregano and cook for 1 more minute Add chiles and sea salt, toss until all is thoroughly heated. Taste and adjust seasoning with salt.
Assembly of Quesadillas: Put two sheets of plastic cut from a plastic storage bag (I use a plastic grocery bag) inside a Tortilla press OR use rolling pin on 2 sheets of wax paper with the dough ball between the 2 sheets. Put a dough ball between the sheets and gently press down the top plate of the press OR roll the dough ball using the rolling pin into a circular shape between sheets of wax paper. Remove top piece of plastic or wax paper and place a generous tablespoon of the shredded cheese on half of the tortilla keeping the edges free. Top with a couple of chile strips and a few onion slices. Lift the lower piece of plastic or wax paper to fold the uncovered side of the tortilla over the filling. Press the edges together with your fingers. Remove from the press or the wax paper and set aside covered with the damp towel. Repeat until all Quesadillas are made.
Cooking of Quesadillas: Preheat oven to 200 deg F. Pour oil to a depth of 1 inch into a deep heavy frying pan and place over medium high heat until the oil shimmers.
Fry the Quesadillas – one at a time until golden 1-2 minutes.
Using a slotted spatula, transfer to paper towels. Let drain briefly then transfer to a heat proof platter and keep warm in the oven. Serve as soon as possible.
I had an extra amount of onion, cheese, and chiles so I dumped it on a store-bought corn tortilla, place another corn tortilla on top and then heated it on a dry frying pan until the cheese melted.