In a food processor, combine the blue cheese, flour, cornstarch, butter, salt and pepper and pulse until the dough just comes together.
Transfer the dough to a sheet of plastic wrap and shape it into a 4-inch disk; wrap and refrigerate until firm, about 1 hour, or for up to 1 day.
Preheat the oven to 375°. Line a baking sheet with parchment paper.
On a well floured surface, roll out the dough to a 12-by-6-inch rectangle about 1/8 inch thick.
Sprinkle the poppy seeds over the dough and press lightly to help them adhere. Cut the dough into 6-by- 1/2 -inch strips and transfer them to the baking sheet; twist the ends in opposite directions to make spirals.
Bake the cheese straws for about 15 minutes, or until crisp and brown. Let the straws cool on the baking sheet before serving.