Make the Creamy Spinach Artichoke Filling: In a medium mixing bowl, blend the softened cream cheese, minced garlic, salt, pepper, and red pepper flakes until smooth. Fold in the thoroughly squeezed spinach, chopped artichoke hearts, mozzarella cheese, and 1/4 cup of the grated Parmesan cheese until evenly combined.
Roll and Spread the Pastry: Lightly flour a clean work surface. Unroll the cold puff pastry sheet and gently roll out any seam lines into a neat 10x14-inch rectangle. Using a spatula, spread the creamy spinach artichoke mixture evenly across the dough, leaving a clean 1/2-inch border along the top long edge.
Roll into a Tight Log: Starting from the long edge closest to you, roll the pastry up snugly into an even log. Brush the clean border lightly with a dab of water or egg wash and pinch the seam to seal. Wrap the log in plastic wrap and chill in the freezer for 15 minutes (or refrigerator for 30 minutes) to firm up so it slices cleanly.
Slice into Pinwheels: Preheat your oven to 400°F (200°C) and line a large baking sheet with parchment paper. Using a sharp chef knife or serrated knife, slice the chilled log into 16 to 18 equal rounds (approx. 3/4-inch thick). Lay the pinwheels flat cut-side up on the baking sheet, leaving 1 inch of space between each.
Egg Wash, Top with Parmesan & Bake: Brush the pastry tops and edges with the beaten egg wash. Sprinkle the remaining 1/4 cup of grated Parmesan over the pinwheels. Bake for 18 to 22 minutes until puffed, crisp, and deep golden brown, and the cheese center is bubbling hot.
Finish and Serve Warm: Remove from oven and immediately brush the warm tops with garlic herb butter. Let settle for 3 minutes, transfer to an entertaining serving platter garnished with rosemary sprigs, and serve warm.