In a food processor, pulse together butter, cheese, egg, mustard, salt, and red-pepper flakes until smooth. Add flour and pulse until combined. Transfer dough to a work surface and form into a 2-inch-wide log. Wrap in plastic and refrigerate until firm, 4 hours (or up to 1 day).
Preheat oven to 350 degrees, with racks in upper and lower thirds. Slice dough into 1/4-inch rounds and arrange, 1 inch apart, on two parchment-lined baking sheets. Bake until edges are golden, 15 to 20 minutes, rotating sheets halfway through. Let cool completely on racks before serving.
Mix It Up: In place of red-pepper flakes, try stirring in 3 tablespoons tomato paste or 1/3 cup crumbled cooked bacon, finely chopped herbs, or finely diced fresh green chiles.
Store crackers in an airtight container, up to 1 day.