Unfold the first sheet of puff pastry on a work surface (if the pastry is sticky, lightly dust the work surface first). Roll it into a 10x12-inch rectangle. Spread with 1/4 cup of pesto leaving a 1/2-inch border on all sides. Lay 5 prosciutto slices on top, overlapping as needed. Sprinkle with 3/4 cup Mozzarella and 2 tablespoons Parmesan.
Starting at the long (12-inch) edge, roll the puff pastry into a log. Press it lightly as you go so that the filling rolls inside the pastry. Trim the ends so that they are even, then tightly wrap the log with the plastic wrap. Place in the refrigerator for at least 20 minutes or up to 1 day. Repeat with the second sheet of puff pastry, and remaining pesto (if using 2 types of pesto, use the second variety here), prosciutto, and cheeses. Preheat your oven to 375 degrees F and line two baking sheets with parchment paper or silicone baking mats.
Once the logs have chilled, using a sharp, serrated knife, carefully cut each log into 1/2-inch-wide slices. (If each log is 12 inches, you will get 24 rolls from each roll—if you are worried about the recipe yield, be sure to measure). Arrange the rolls on the baking sheet spiral-side up, spacing each 1 inch apart. Bake until the puff pastry is flaky and golden brown and the cheese is hot and bubbly, about 18 minutes. Immediately sprinkle with chopped basil as desired. Let cool slightly. Serve warm or at room temperature.
TO MAKE AHEAD AND FREEZE: Fully bake the pinwheels, then let cool. Arrange in a flat layer on the baking sheet and freeze completely. Transfer to an airtight container and store for up to 1 month.
You can also freeze the entire unbaked log. Prepare your log and wrap it very tightly with plastic wrap, then store in the freezer for up to 2 months. Let thaw overnight in the refrigerator, and slice and bake as directed.
You could also try freezing the pinwheels unbaked, but be careful of the grated cheese coming loose. Baking the pinwheels first, then freezing, is the easiest route.
TO REHEAT: Reheat directly from frozen on a parchment lined baking sheet at 375 degrees F until warmed through (this takes just a few minutes).