Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper and set aside.
Unfold the puff pastry on a lightly floured surface. Roll into a 10-inch x 14-inch rectangle, then carefully transfer to the prepared baking sheet. In a small bowl, beat together the egg yolk with 1 tablespoon water to create an egg wash (save the egg white for a different use). Lightly brush the entire surface of the puff pastry with the egg wash. With a sharp knife, score a border around the pastry dough that is 1 inch from the edges on all sides, so that you mark a rectangle. With a fork, pierce holes in the pastry all over the center of the rectangle you just created (do not pierce the edges). Each fork imprint should be about 1/2-inch apart. Bake the puff pastry for 10 minutes. It will be golden and slightly puffed. If any areas are very puffy, gently press them down with a fork.
Meanwhile, in a small bowl whisk together the ricotta, 1 whole egg, lemon zest, lemon juice, parsley, 1/4 cup Parmesan, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Spread in an even layer over the crust, staying inside the rectangle and leaving the 1-inch border you created uncovered. Trim the bottoms of the asparagus spears to fit crosswise inside the rectangle (about 8-inches in length). Arrange them in a single layer over the ricotta, alternating ends and tips. Brush the asparagus lightly with the olive oil, and sprinkle with the remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Tear the prosciutto into pieces and scatter it over the top.
Bake for 15 minutes, then check to make sure the pastry isn't browning too quickly. If it is, lightly tent the tart with foil, then return to the oven and continue baking until the asparagus is tender, about 5 to 7 additional minutes. Remove the tart from the oven and let cool for 2 minutes. Slice and serve, sprinkled with additional Parmesan as desired.