Place the flour, salt and sugar into a medium bowl and whisk to combine. In a small bowl whisk together the water, egg, and 2 tablespoons of the butter. Add the egg mixture to the dry ingredients and stir until the dough comes together, adding additional water if the dough is dry. Knead the dough in the bowl 5 to 6 times. Turn the dough out onto the counter, divide into thirds, cover with a tea towel and allow to rest for 30 minutes.
Preheat the oven to 375 degrees F.
Lightly butter the back of a half sheet pan using some of the remaining butter. Place the sheet pan, upside down on a surface that will prevent sliding. Working with one ball at a time, place the dough ball onto the back of the sheet pan and roll the dough out to an even 1/8-inch thickness. Gently stretch the edges of the dough so they fall slightly below the edge of the pan and hold the dough in place. Lightly brush the dough with butter, place on the middle rack of the oven and bake for 10 to 15 minutes or until golden brown. Remove the lavash to a cooling rack. Repeat with the remaining dough, on a cooled pan. Break each sheet into shapes and sizes as desired. Once completely cooled, store in an airtight container for up to 5 days.