Preheat oven to 400 degrees F. Lightly coat 2 baking sheets with cooking spray or line with parchment paper.Advertisement
Whisk together egg white, oil, salt and cayenne, if using, in a small bowl.
Lay a sheet of phyllo on a work surface with a short side toward you. Keep remaining phyllo sheets covered with plastic wrap or wax paper. With a pastry brush, lightly coat the lower half of the sheet with the egg-white mixture and sprinkle with about 2 teaspoons cheese. Fold the upper half over to cover the lower half. Brush the right half of the folded sheet with egg-white mixture, sprinkle with a heaping teaspoon of cheese and fold the left half over the cheese. Brush the bottom half of the folded sheet with the egg-white mixture, sprinkle with 1 teaspoon cheese and fold the upper half over the lower half. Finally, brush the top with the egg-white mixture and sprinkle with 1 teaspoon cheese. Cut into ten 1/2-inch strips using a knife or serrated pastry cutter. With a wide spatula, transfer the strips to the prepared baking sheet, placing them about 1/2 inch apart. Repeat the procedure with the remaining 5 sheets of phyllo, egg-white mixture and cheese.
Bake the straws until golden and crisp, 8 to 10 minutes. Transfer to a rack to cool.
Make Ahead Tip: Store in an airtight container at room temperature for 1 week or freeze for up to 2 months.