**Preheat the Oven:**
Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.
**Prepare the Puff Pastry:**
Roll out the puff pastry sheet on a lightly floured surface into a rectangle, about 10x12 inches. If desired, spread a thin layer of Dijon mustard over the surface for extra flavor.
**Add the Filling:**
Sprinkle the Gruyère and Parmesan cheese evenly over the pastry, leaving a small border around the edges. Top with the chopped parsley and green onions. Season lightly with salt and pepper.
**Roll and Slice:**
Starting from one long edge, tightly roll the puff pastry into a log. Use a sharp knife or pizza cutter to slice the log into 1/2-inch thick pinwheels.
**Brush with Egg Wash:**
Arrange the pinwheels on the prepared baking sheet, spacing them about 1 inch apart. Brush the tops and sides of each pinwheel with the beaten egg.
**Bake:**
Bake for 15–18 minutes, or until the pinwheels are golden brown and puffed. Remove from the oven and let cool slightly before serving.