Preheat the oven to 400 degrees . Line a large baking sheet with parchment. Butter a large baking sheet.
In a medium bowl, combine the flour and grated butter. Add the cheeses and mix, using your hands, until well combined. Stir in the water. On a clean, floured work surface, knead the dough until smooth, sprinkling in a few drops of ice water if the dough is too dry. Shape into a disk.
Roll out the disk to form a large rectangle (about 9 by 12 inches and 1/4 inch thick). Cut 1/2-inch strips from the longer side, then cut those in half to form 6-inch strips. Sprinkle the cheese straws with salt, transfer to the baking sheet with a spatula and bake until golden brown, about 15 minutes.