Split the kielbasa in half lengthwise and then cut each half into six 3/4-inch pieces, slicing on an angle. Cook the pieces in a large nonstick skillet over medium heat until the skin is crisped and some of the fat has been released, about 5 minutes. Transfer the kielbasa to a paper towel-lined plate to cool.

Preheat the oven to 425 degrees F.

On a lightly floured surface, roll one piece of puff pastry out to form an 8- by 11-inch rectangle. Using a 3 1/2-inch round cookie cutter, cut out 6 circles from the pastry; cut the circles out as close together as possible and reserve the scraps. Repeat with the second sheet of puff pastry.

Wrap a circle of puff pastry around each of a kielbasa, pinching the seams together to seal all the way around so no kielbasa is exposed and gently pulling on the ends to make a tapered football shape. Place on an ungreased baking sheet, seam-side down, and repeat with the remaining pastry and sausage.

Brush the rolled sausages with the beaten egg. Use bits of the pastry scraps to make laces on each "football" by placing 1 strip lengthwise on top of each and laying 3 smaller strip crosswise over it. Press down gently to adhere and brush the laces with a bit more egg.

Bake until the pastry is browned and crisped, 25 to 30 minutes. Serve the pigskins with mustard for dipping.