Preheat the oven to 180°C. Brush two flat baking trays with melted butter and line with nonstick baking paper. (The butter will prevent the paper from moving around on the tray.)
In a mixing bowl, combine the cheese and butter. Blend in the flour and salt. Form the mixture into 6 balls. Roll the balls into slender "snakes." Cut each "snake" into 4 straws. Bake the straws for about 8 minutes, or until golden brown.
Cream the butter and the Cheddar and Parmesan cheeses together in a medium bowl until well combined. Add the flour, paprika, cayenne pepper, salt and water and mix with a wooden spoon until the dough comes together in a ball. Turn the dough out on a lightly floured surface and press into a square. Wrap the dough in greaseproof paper and place in the refrigerator for 1 hour to chill.
Roll the dough out between two sheets of nonstick baking paper into a 24 x 20 cm rectangle 5 mm thick. Trim the edges of the pastry. Cut the dough into straws 10 cm long by 15 mm wide. Place the straws on the prepared trays, allowing a little room between each straw. Refrigerate the trays for 10 minutes, or until the dough is firm.
Brush the cheese straws with lightly beaten egg and bake for 10–12 minutes, or until the straws are light golden and firm to the touch. (Halfway through the cooking time, swap over the top and bottom trays in the oven.) Allow the cheese straws to cool on the baking trays for 5 minutes before transferring them to a wire rack to cool completely.