Bring a large pot of salted water to a boil. Blanch the asparagus for 2 minutes, then immediately transfer to an ice bath to stop the cooking. Drain well and pat completely dry.
Roll out the puff pastry on a lightly floured surface and cut it into 8 wide strips. The strips should be wide enough to wrap around the asparagus bundle with a slight overlap.
Working one at a time, wrap 3 asparagus spears diagonally with a slice of prosciutto.
Lay a slice of provolone on the work surface and place the prosciutto-wrapped asparagus bundle on one end. Roll the cheese around the bundle.
Place the bundle at one end of a strip of puff pastry and roll it diagonally until completely enclosed. Arrange the seam-side down on a parchment-lined baking sheet.
Brush each bundle generously with the beaten egg.
Refrigerate the assembled bundles for 15–20 minutes to allow the puff pastry to firm up. This helps the pastry hold its shape and reduces cheese from leaking during baking.
Bake in a preheated 400°F oven for 20–25 minutes, or until the puff pastry is deeply golden brown and crisp.
Let cool for 3–5 minutes before serving.