Whisk the flour, salt, and sugar together in a medium bowl. In a separate small bowl, whisk together the water, egg, and 2 tablespoons of the meltd 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 rest for 30 minutes.
Heat oven to 375ºF.
Place a half-sheet pan upside down on a surface that will prevent sliding and grease using some of the remaining butter. Working with one ball at a time, place the dough onto the back of the sheet pan and roll 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 until golden brown, 10 to 15 minutes. Remove the lavash to a wire rack to cool. Repeat with the remaining dough, using a cooled half-sheet pan. Break each sheet into shapes and sizes as desired. Once completely cooled, store in an airtight container for up to 5 days.