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, 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, 2 or 3 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.

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 squares about 1-inches long. Use the back end of a toothpick to make hole.

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.