Combine flour, butter, and cayenne pepper in a bowl and rub together with your fingers until the mixture resembles bread crumbs. Mix in Parmesan cheese. Add whole egg, egg yolk, and mustard and mix until dough comes together.
Tip dough onto a lightly floured work surface and knead lightly for 30 seconds. Wrap dough in plastic wrap and chill for 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Grease 2 baking sheets.
Roll out dough with a floured rolling pin on a floured work surface into a 9x12-inch rectangle. Cut with a sharp knife or pizza cutter into 24 straws and carefully twist each straw twice. Lay the straws onto the prepared baking sheets. Beat egg white with a fork until frothy and brush over cheese straws. Sprinkle with sesame and poppy seeds.
Bake in a preheated oven until golden, 18 to 20 minutes.