Preheat oven to 350°F. Line a baking sheet with a silicon mat or sheet of parchment. To the jar of a food processor, add the grated cheese, butter, flour, salt, paprika and cayenne. Pulse until the mixture is combined. Add the milk and pulse for about 30 seconds, until the dough forms a ball.
On a lightly floured piece of parchment, using a floured rolling pin, roll the dough to 1/8-inch thickness. Using a pasta cutter or a pizza cutter, cut 1 by 1-inch squares. Using the end of a digital thermometer or blunt side of a skewer, make holes in the center of each cheez-it.
Transfer the crackers to the parchment-lined baking sheet. I found using a sharp paring knife very helpful in this transfer. Place in the oven to bake for 12 to 15 minutes, or until the ends are barely browned. While the crackers are baking, melt 1 tablespoon of butter in a small saucepan, transfer to a small ramekin. In a teeny bowl, mix together the remaining 1/2 teaspoon of paprika, pinch of cayenne pepper and a few pinches of salt. Remove from the cheez-its from the oven and immediately brush the crackers with a teeny bit of melted butter and then top with a light sprinkling of paprika/cayenne/salt mixture. Allow to cool on the baking sheet.