4 tablespoons unsalted butter (1/2 stick), at room temperature
2 teaspoonskosher salt
1 teaspoon granulated sugar
2 1/4 teaspoons coarsely ground black pepper
Whisk together egg and water until evenly combined; set aside. Heat the oven to 400°F and arrange the rack in the middle.
Combine flour, butter, salt, and sugar in the bowl of a stand mixer fitted with the paddle attachment. Mix on low speed until ingredients resemble cornmeal, about 2 minutes. Add egg mixture and mix on medium-low speed until just incorporated. Divide into three pieces, form into flat disks, cover each in plastic wrap, and refrigerate at least 30 minutes.
Place one dough piece on a lightly floured surface and, using a floured rolling pin, roll dough into a paper-thin sheet (about 1/16 inch thick), approximating the dimensions of your baking sheet.
Place dough on an ungreased baking sheet and sprinkle with 3/4 teaspoon of the pepper. Press pepper into lavash and bake until golden brown and crisp, about 18 minutes. Meanwhile, prepare remaining dough. Once baked, transfer to a wire rack to cool. Serve warm or at room temperature.