How to Make It
In the bowl of a standing mixer fitted with a dough hook, dissolve the yeast in the water. Stir in 2 cups of the flour and the honey and brown sugar. Beat on medium speed for 1 minute. Cover with a kitchen towel and let stand for 1 hour.
On low speed, gradually beat the remaining 1 1/4 cups of flour and the salt into the sponge. Knead the dough at medium speed for 5 minutes. Cover and let stand at room temperature until doubled in bulk, about 2 hours.
Preheat the oven to 475°. Lightly oil 3 large baking sheets. In a small bowl, combine the poppy seeds and sesame seeds. Transfer the dough to a lightly floured surface and shape into a log. Cut the dough into 6 equal pieces. Working with 1 piece at a time and keeping the rest covered and refrigerated, on a lightly floured work surface, roll out the dough round to a 14-by-9-inch rectangle; if the dough seems too elastic, let it relax for a minute. Transfer the rectangle to an oiled baking sheet, brush lightly with the beaten egg and sprinkle with 1 1/2 teaspoons of the seeds.
Bake the lavash in the upper third of the oven for 8 to 10 minutes, or until crisp and golden brown in patches. Slide the lavash onto a rack and let cool completely. Meanwhile, roll out and bake the remaining lavash, oiling the baking sheets as necessary. Serve the lavash, breaking it into pieces at the table.