Roughly chop the Kadayıf into 1 inch (2.5 centimeter) pieces and place in a large bowl.
Pour the melted butter over the chopped Kadayıf and toss to thoroughly coat.
Press a layer of the coated Kadayıf across the bottom and sides of the prepared pans.
Cover each layer of pastry with a layer of the cheese, leaving about a 1/2 inch (1.25 centimeter) border around the edges.
Cover the cheese with another layer of the Kadayif. Press down to completely enclose the layer of cheese, especially around the sides.
Place the pan over medium low heat. Cook until the bottom is golden, adjusting the heat between medium and low as needed, for about 6 minutes.
Carefully flip the pastry onto a plate.
Grease the pan again with the grape molasses butter mixture and place the pastry back into the pan, golden side up.
Place the pan back over medium low heat and cook until golden on the bottom, another 4-5 minutes.
Remove the pan from heat and immediately pour the desired amount of cooled syrup over the pastry to soak.
Serve the Künefe hot while the cheese is still melted and stretchy.