Line a baking sheet with aluminum foil and lightly grease with vegetable oil.
Place butter into a mixing bowl; add Cheddar cheese, Parmesan cheese, paprika, cayenne pepper, and salt. Mix together with the back of a spatula until thoroughly combined.
Mix flour into cheese mixture with a fork until crumbly. Sprinkle in cold water, 1 or 2 drops at a time, and mix with spatula until it comes together in a dough that holds its shape when squeezed.
Transfer dough to a work surface and press into a thick, flattened disk. Wrap in plastic wrap and refrigerate 30 minutes.
Preheat oven to 375 degrees F (190 degrees C).
Roll dough out on a floured work surface to about 1/8-inch thick. Use a pizza cutter to cut 1-inch wide strips from the dough. Cut strips crosswise into rectangles about 1 1/2-inches long. Use the back of a bamboo skewer to punch 5 small holes into each cracker. Arrange crackers onto prepared baking sheet.
Bake in the preheated oven until crackers are browned and crisp, about 15 minutes. Let stand about 3 minutes for crackers to cool and release from the foil. Let cool completely before serving.
