About Us. Cuba Cheese History to The Cuba Cheese Shoppe Specialty Grocery Store in Cuba, New York. People talk about spicy cheese curds, horseradish cheese and wisconsin cheddar. See reviews and. Contact Us. Cuba Cheese Shoppe. 53 Genesee Street. PO Box Cuba, NY. Discover Allegany County: Cuba Cheese Shoppe Choose from a variety of Cuba Cheese Gourmet Gifts! We can ship anywhere. My son was visiting from Georgia and I took him to the shop, because he loves cheese. We enjoyed testing the different kinds. They also have sauces, jellies. Cuba Cheese Shoppe, $ Off Purchase of $ or more. Jun 10, Cuba Cheese Shoppe Restaurants Ads from Cuba Pennysaver Cheese Shop.