Sprinkle 1/4 cup of the cheese over half of 1 tortilla. Then place 3/4 cup coleslaw mix on cheese. Drizzle about 3 tablespoons of the salsa mixture over coleslaw. Lay 1/4 of the corned beef onto coleslaw; top with 1/4 cup more cheese. Fold tortilla over to cover filling. Repeat to make 3 more quesadillas.
Place quesadillas slightly apart on baking sheets. Broil 4 to 6 inches below heat until crisp and lightly browned, 1 to 2 minutes (watch closely). Using a wide spatula, carefully turn each quesadilla over. Broil until crisp and lightly browned on other side, 1 to 2 minutes more.
Cut quesadillas into wedges. Serve with remaining salsa mixture.