Turn your griddle to medium-low heat, around 300-325°F. Let the griddle preheat for about 5 minutes so the temperature evens out across the whole surface.
Once the griddle is up to temp, add a small amount of butter or oil, about 0.5 tablespoon per quesadilla. Spread it around with your spatula so you have a thin, even layer.
Place your first tortilla flat on the buttered section of the griddle. Add about 0.75 cup of shredded cheese per quesadilla directly onto the tortilla, spreading it evenly and leaving about half an inch around the edges.
If adding other fillings, put them on top of the first layer of cheese. Use about 0.25 cup of cooked chicken or a few tablespoons of peppers and onions per quesadilla.
Place the second tortilla on top and press down gently with your spatula.
After about 2-3 minutes, use your spatula to lift an edge and check the bottom. When the bottom is golden brown, flip the quesadilla in one confident motion.
Cook the second side for another 2-3 minutes, pressing down on the quesadilla with your spatula a few times to help everything compress and meld together.
The quesadilla is done when both sides are golden brown and you can see melted cheese around the edges.
Transfer the quesadilla to a cutting board and let it sit for about 1 minute before cutting.
Cut into quarters or sixths with a pizza cutter or sharp knife. Serve immediately with desired toppings.