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.
The lavash can be wrapped well and kept at room temperature for 1 day.