Combine flour, cornmeal, salt, cayenne, and nutmeg in bowl of a food processor fitted with the metal blade; pulse to combine. Add butter, and pulse until mixture resembles coarse meal. Add cheese, and pulse until combined. With machine running, add milk. Process until dough comes together and is well combined.
Transfer dough to a clean work surface. Shape dough into a 1 3/4-inch-wide log. Wrap with 2 layers plastic wrap and refrigerate for at least 24 hours and up to 2 weeks or frozen for up to 4 weeks.
Preheat oven to 325 degrees. Slice well-chilled log into 1/4-inch-thick slices. Transfer the slices to a parchment-lined baking sheet, and bake immediately, rotating sheet halfway through baking, until crackers are golden brown and firm in the center, 25 to 35 minutes. (Crackers should not get too dark around edges.) Transfer to a rack to cool. Crackers may be made a day ahead and kept in an airtight container at room temperature.