Monterely jack cheese may be authentic as a US South-Western version but you can use any melting cheeses such as Cheddar, Gruyere etc. I happened to have a “Swiss” style cheese similar to Raclette with interesting herbal flavors. I usually add seeded and finely diced jalapeño pepper but I did not have one so I used canned roasted mild green pepper but you do not have to use any pepper. I added a small amount of vegetable oil to a frying pan and placed a tortilla (I used wheat tortilla). I then put shredded cheese and the pepper on half of the tortilla. While the tortilla was still pliable, I folded it in half and pressed it with a spatula then turned it over as seen below. I cooked it until the tortilla was browned and crispy and the cheese was melted.
Like some favorite Japanese dishes may have an origin in China, many so-called South-Western cuisines in the U.S. are derived from or influenced by Mexican cooking. Quesadilla is one of these and a perfect bar food. Essentially, a cheese sandwich using tortillas. There is no real recipes. You can make many variations depending on types of cheeses and other proteins such as cooked shredded chicken, pork etc. My Mexican friend tells me that the authentic quesadilla is made from one tortilla and folded in half into a half-moon shape but using two tortillas may be more efficient. I just wanted to make a small amount for two of us this time, so I made it from one tortilla.