Get out 1 tortilla on a work surface. Place about a quarter of the cheese on one half, not too near edge so cheese will not melt out as it cooks.
Top cheese with a quarter of the avocado and four jalapeno slices. Fold tortilla in half over filling to form a semicircle.
Heat a grill pan or skillet over medium heat. Slide uncooked quesadilla onto the pan. Squish down with a weight or just press briefly with a spatula. Cook 1 minute, then flip the quesadilla and grill 1 minute more or until cheese is melted.
Transfer the quesadilla to a cutting board and slice into 6 wedges. Serve immediately. Repeat with remaining ingredients. Makes 24 wedges.