Generously dust a work surface with flour. Place the pastry on the prepared work surface, dust lightly with flour, and, using a rolling pin and sharp knife roll and trim it into a 10-by-12-inch rectangle.
Using a pastry brush, coat the surface of the pastry evenly with the pesto. Arrange the prosciutto slices in a single layer, overlapping slightly. Sprinkle the cheeses evenly over the prosciutto. Roll pastry into a cylinder – wrap and freeze until firm, at least 10 minutes.
Meanwhile, heat oven to 400°F and arrange the rack in the middle. Line a baking sheet with parchment paper and set aside. Using a serrated knife, trim each end then cut the roll crosswise into ½ -inch slices. Arrange the slices on the prepared baking sheets about 1 inch apart. You should have about 15 pinwheel bites.
Bake until the pinwheel bites are lightly browned, about 8 to 10 minutes. Remove the baking sheets from the oven and flip the pinwheels over with a flat spatula. Return to the oven and bake until golden brown, about 5 minutes more. Remove the pinwheels to a wire rack to cool slightly then serve.