I used mozzarella cheese because it's what I had (and it melts so nice), but monterey jack would probably work just as good.
There are no specific measurements because it's a use-what-you've-got deal; more a great idea than a recipe.
Cut up the corned beef, potatoes, carrots, and cabbage so they are small and uniform in size.
Heat a large non-stick skillet over medium to medium-high heat (I don't grease my pan). Layer a flour tortilla, 3-4 tablespoons mozzarella cheese, the cut up corned beef, carrots, potatoes and cabbage, another 3-4 tablespoons mozzarella cheese, and another tortilla.
Cook the quesadilla until it's golden brown. Flip it over (carefully so the filling doesn't fall out) and cook it on the other side until golden brown.
Cut into wedges and enjoy :)