Preheat the oven to 400°F. Line 2 baking sheets with parchment paper. Roll each sheet of pastry into an 11 x 14 inch rectangle. Brush one side of each sheet with the egg wash.
Sprinkle the cheese evenly over one of the egg-washed pieces of pastry. Season well with black pepper. Place the second piece of pastry, egg side down, on top of the cheese. Run the rolling pin over the pastry sheets to seal.
Using a sharp knife, cut the pastry crosswise into 32 strips. If the pastry gets too warm and stretchy at any time, chill it for 15 minutes before proceeding.
Gently twist each strip 3 to 4 times and arrange on the baking sheets. Brush the twisted strips with the egg wash, then sprinkle with the sesame and poppy seeds.
Bake until the cheese straws are golden brown, about 15 minutes. Remove from the oven and serve warm or at room temperature.
Note: Any unbaked cheese straws can be frozen on their baking sheets, then transferred to resealable plastic bags, and kept in the freezer for up to 2 weeks.