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 for 30 minutes.

Preheat oven to 375°F (190°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.