Using stand mixer fitted with dough hook, mix semolina flour, whole-wheat flour, all-purpose flour, and salt together on low speed. Gradually add water and oil and knead until dough is smooth and elastic, 7 to 9 minutes. Turn dough out onto lightly floured counter and knead by hand to form smooth, round ball. Divide dough into 4 equal pieces, brush with oil, and cover with plastic wrap. Let rest at room temperature for 1 hour.

Adjust oven racks to upper-middle and lower-middle positions and preheat oven to 425 degrees. Lightly coat two 18 by 13-inch rimless (or insulated) baking sheets with vegetable oil spray.

Working with 2 pieces of dough (keep remaining dough covered with plastic), press dough into small rectangles, then transfer to prepared sheets. Using rolling pin and hands, roll and stretch dough evenly to edges of sheet. Using fork, poke holes in dough at 2-inch intervals. Brush dough with beaten egg, then sprinkle each with 1½ teaspoons sesame seeds, ½ teaspoon salt, and ¼ teaspoon pepper. Press gently to help seasonings adhere.

Bake crackers until deeply golden brown, 15 to 18 minutes, switching and rotating sheets halfway through baking. Transfer crackers to wire rack and let cool completely, then break into large crackers. If not serving immediately, let baking sheets cool completely before rolling out remaining dough. Break cooled lavash into large crackers and store at room temperature for up to 2 weeks.