Preheat oven to 400°F In a large saucepan, melt butter. Add 2 tablespoons sugar and stir until smooth. Add apple and cook on medium-low heat until pieces are slightly softened but still hold their shape, 5-7 minutes. Stir in 1/8 teaspoons cinnamon. Transfer to a bowl adn set aside.
In the same saucepan, add apple cider, cider vinegar, and remaining sugar and cinnamon and heat on medium until sugar dissolves. Turn heat to low and simmer until cider mixture is a syrupy and reduced to about 3/4 cup approximately 25 minutes.
Meanwhile, on a lightly floured surface, roll out puff pastry into a 14x14-inch square. Use a knife or pizza wheel to cut pastry into 12 rectangular pieces by slicing 4 horizontal rows and 3 vertical rows. Place cut pastry pieces into a greased 12-cup muffin tin, lining bottoms and sides. Use a fork to prick the bottoms a few times. Spoon cooked apple into pastry cups. Bake 15 minutes or until golden brown. When cool enough to handle, remove the tartlets from the muffin tin, drizzle with cider syrup and sprinkle with cheese. Serve hot or cold.