In the bowl of an electric mixer fitted with the paddle attachment, cream the butter and Stilton together for 1 minute, or until smooth. With the mixer on low speed, add the flour, salt and pepper and mix until it's in large crumbles, about 1 minute. Add 1 tablespoon of water and mix until combined.
Dump the dough onto a floured board, press it into a ball, and roll into a 12-inch long log. Brush the log completely with the egg wash. Spread the walnuts in a square on a cutting board and roll the log back and forth in the walnuts, pressing lightly, and distributing them evenly on the outside of the log. Wrap in plastic and refrigerate for at least 30 minutes or for up to 4 days.
Meanwhile, preheat the oven to 350 degrees F.
Cut the log 3/8ths-inch thick with a small, sharp knife and place the crackers on a sheet pan lined with parchment paper. Bake for 22 minutes until very lightly browned. Rotate the pan once during baking. Cool and serve at room temperature.