Mix together flours, shortening, salt, and sugar in mixer using dough hook attachment at medium speed until mixture resembles coarse meal. Add water and egg and mix 5 minutes (dough will be very sticky). Scrape down side of bowl and cover with plastic wrap, then let stand at warm room temperature at least 1 and up to 4 hours.
Put oven racks in upper and lower thirds of oven and preheat oven to 350°F.
Stir together butter (2 tablespoons) and garlic. Lightly brush baking pans with additional melted butter.
Turn out dough onto a well-floured surface and dust generously with flour, then divide dough into 2 equal portions. Roll out 1 portion of dough with a lightly floured rolling pin into a roughly 10- by 7-inch rectangle. (Dough will be very soft.) Dust backs of your hands with flour and slide them under dough. Working over 1 of the baking pans, gently stretch dough as evenly as possible to fit into pan. Pull sides and corners of dough to cover bottom of pan evenly. (Sides of dough may be slightly thicker than center.) Repair any holes in dough by pinching and pressing dough together. Repeat with second piece of dough and other baking pan.
Brush surface of dough lightly with garlic butter (be careful not to tear the dough), then sprinkle evenly with sesame seeds, lightly pressing to help them adhere. Sprinkle with cheese.
Bake, switching position of baking pans in oven and rotating them 180 degrees halfway through baking, until golden, 25 to 35 minutes total.
Cool lavash in pans on racks, about 15 minutes. (Lavash will crisp as it cools.) Break into pieces to serve.
Cooks' note: Lavash can be cooled completely and kept in an airtight container at room temperature 3 days.