In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and cheese together until smooth, about 1 minute. With the mixer on low, add the flour, salt, and pepper and continue beating for about 1 minute, until the dough is in large crumbles. Add 1 tablespoon cold water and mix until the dough comes together. If necessary, add a little more water, a few drops at a time. Dump the dough onto a floured work surface and roll it into a 12-inch-long log.
Spread the walnuts on a cutting board. Lightly brush the surface of the log with the egg wash, turning to coat all sides. Roll the log back and forth in the walnuts, pressing lightly, to distribute them evenly all over the surface of the dough. Wrap in plastic wrap and refrigerate for at least 30 minutes or up to 4 days.
Meanwhile, preheat the oven to 350°F (176°C). Line a baking sheet with parchment paper.
Use a small sharp knife to cut the log into 3/8-inch-thick slices and arrange the crackers on the parchment. Bake for 22 minutes, until very lightly browned, rotating the sheet once during baking. Let the crackers cool on the baking sheet. Serve at room temperature.