This recipe adapted from "Martha Stewart's Hors d'Oeuvres Handbook" (Clarkson Potter, 1999).
Combine flour, salt, pepper, and thyme in the bowl of a food processor; pulse to combine. Add butter; pulse until mixture resembles coarse meal. Add cheese; pulse until combined. With machine running, add the milk; process until dough comes together and is well combined.
Transfer dough to a work surface. Shape dough into a 2-inch-wide log. Wrap with plastic wrap, and refrigerate for at least 24 hours.
Heat oven to 325 degrees. Slice chilled log into 1/4-inch-thick slices. Transfer slices to a parchment-lined baking sheet. Dip a sprig of thyme into egg white, and place in center of a cracker slice; repeat with remaining thyme and crackers. Bake immediately, rotating sheet once, until crackers are golden brown and firm in the center, 25 to 35 minutes. Transfer to a rack to cool.
Crackers may be made a day ahead and kept in an airtight container at room temperature.