Place the butter, grated cheddar, salt, white pepper, and cayenne in the bowl of a large food processor. Pulse until butter breaks up into small pieces and the mixtures resembles cottage cheese. Add the flour to the food processor and pulse until the mixture resembles damp breadcrumbs. Add the Worcestershire and pulse a couple more times. The dough will be moist but it should come together in your hands.
Turn the dough out onto the counter or a silicone mat. Knead it a couple of times to form a ball. Shape into a disk, wrap in plastic wrap, and place in the refrigerator to chill for 1 hour.
Place the oven rack in the center position of the oven and preheat to 350°F. Line a large sheet pan with parchment paper. Remove the dough from the refrigerator and very lightly flour your work surface. Roll the dough into a rectangular shape measuring 1/8-inch thick (if the dough seems sticky, you may use a bit of flour on your rolling pin, but it probably won't be necessary). Using a pizza cutter, pastry wheel, or sharp knife, cut the dough vertically into approximately 1-inch wide strips, then cut across the strips horizontally to make 1-inch squares (you may use a clean ruler as a guide, if you wish). Form a ball with any leftover scraps of dough, roll out again, and repeat until all crackers are cut out.
Transfer crackers to baking sheet. The crackers shouldn't spread, so it's not necessary to space them out much. Use a wooden skewer to poke a small hole in the center of each cracker. Bake for 15 minutes or until crackers are just starting to darken around the edges. Transfer crackers to a cooling rack and allow to cool completely before storing in an airtight container.