Wash, core and slice apple into quarter inch slices. Wash and roughly chop arugula.
Melt butter in pan and add apple slices and arugula. Cover. Sautée over medium heat until apples are soft and arugula is wilted, about 10 minutes.
While the apples and arugula cook, grate gruyere cheese and set aside
To assemble, arrange two tortillas on a cutting board. Sprinkle each with half the gruyere cheese, then top with ham, arugula and apples. Sprinkle with remaining cheese and top with the remaining two tortillas.
Heat pan with 1 teaspoon of oil and cook one quesadilla at a time until tortillas are golden brown and cheese is melted. Repeat with the other teaspoon of oil and quesadilla.
Cut into wedges and eat immediately. Serves two.