Preheat your oven to 400°F.
In a skillet, melt the butter over medium heat. Add the chopped onion and sauté until soft and caramelized, about 10-12 minutes.
Stir in the garlic powder, thyme, salt, and pepper. Remove from heat and let cool slightly.
In a large bowl, combine the ground sausage, sautéed onion mixture, and shredded mozzarella cheese. Mix until well combined.
On a lightly floured surface, roll out one sheet of puff pastry to smooth out creases. Cut the pastry into rectangles, approximately 4 inches by 6 inches.
Place a generous tablespoon of the sausage mixture along one edge of each rectangle. Roll the pastry over the filling, sealing the edges with a fork. Repeat with the second sheet of pastry.
Place the rolls on a baking sheet lined with parchment paper. Brush the tops with the beaten egg for a golden finish.
Bake for 20-25 minutes or until the pastry is puffed and golden brown.
Let the sausage rolls cool for a few minutes before serving. They can be enjoyed warm or at room temperature.