Preheat the oven to 350 degrees F (175 degrees C). Cut 2 sheets of parchment paper to fit on a baking sheet.
Place 1 sheet of parchment on a flat work surface. Gently place 1 sheet of phyllo dough on top. Brush lightly with butter right up to the edges, being careful not to tear the sheet. Place another sheet of phyllo on top. Press down to remove any air pockets, and brush with butter. Repeat 3 more times. On the 5th sheet, brush with egg white instead of butter. Sprinkle with 1/2 of the bagel seasoning.
Cut the phyllo into 40 squares, or into the shape of your choice, using a pizza cutter. Slide the whole parchment sheet onto the baking sheet.
Bake in the preheated oven until lightly browned, 6 to 8 minutes. Keep a close eye on them, as they tend to brown quickly. Remove parchment sheet from baking sheet and let cool.
Repeat process with remaining ingredients. Let cool and serve immediately, or store at room temperature in a sealed container.