Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a mixing bowl, combine the sausage meat with dried thyme (if using), salt, and pepper. Mix well until evenly combined.
On a lightly floured surface, roll out the puff pastry sheet to approximately ⅛ inch (3 mm) thickness.
Cut the rolled pastry into rectangles, about 4 inches (10 cm) wide and 6 inches (15 cm) long.
Place a generous amount of the sausage mixture along one edge of each rectangle, leaving about half an inch (1.5 cm) border.
Roll the pastry over the sausage filling to form a log shape, sealing the edge with a little beaten egg.
Place the sausage rolls seam-side down on the prepared baking sheet.
Brush the tops of the rolls with the beaten egg to give them a golden color when baked.
Bake in the preheated oven for about 25 minutes or until golden brown and puffed up.
Once baked, remove from the oven and allow to cool slightly before serving.