1. In a small heatproof bowl, soak the figs in boiling water until softened, about 20 minutes. Drain the figs and pat dry. Slice off the tough stems, then cut the figs into 1/2-inch dice.
2. In a food processor, combine the flour, butter, Cheddar cheese, kosher salt and pepper and pulse until the mixture resembles coarse meal. Scrape the mixture into a large bowl and add the figs and egg. Using your hands, work the dough until it comes together.
3. Scrape the dough onto a large piece of plastic wrap and pat it into a 12-by-2-inch rectangular log. Wrap the log tightly and refrigerate until firm, at least 1 hour, or overnight.
4. Preheat the oven to 375°. Butter 2 large baking sheets. Slice the log of dough crosswise 1/4 inch thick. Arrange the slices at least 1 inch apart on the prepared baking sheets. Bake the Cheddar crackers for about 12 minutes, or until they are golden brown and slightly firm. Carefully transfer the crackers to a wire rack to cool.
Make Ahead: The dough can be tightly wrapped in plastic and refrigerated for up to 2 days or wrapped in plastic, then in aluminum foil and frozen for up to 1 month. The crackers can be stored at room temperature in an airtight container for up to 2 days.
Wine Recommendation: A fruity white California sparkling wine will highlight the sweetness of the dried figs and contrast with the salty, creamy, sharp Cheddar cheese in these crackers. Two good examples: Nonvintage Domaine Chandon Brut Classic and Nonvintage Mumm Cuvée Napa Brut Prestige.