Preheat the oven to 400° and line a large rimmed baking sheet with parchment paper.
On a lightly floured sheet of parchment paper, gently roll out the puff pastry to a large rectangle (about 10-by-15-inches). Using a paring knife, gently score a ½-inch border around the pastry. Do not cut through the dough. Using a fork, prick the puff pastry all over within the ½-inch border. Transfer the pastry to the prepared baking sheet. If possible, return to the refrigerator or freezer for 10 minutes. Bake the puff pastry for about 12 minutes, or until beginning to brown.
Meanwhile, in a medium skillet, heat 1 tablespoon of the olive oil. Add the asparagus and cook over moderately-high heat, stirring occasionally for 2 minutes. Add the garlic and lemon juice and season with salt and pepper. Scrape the asparagus into a bowl.
Remove the puff pastry from the oven. Using your paring knife, gently score along the pre-cut border and then lightly press on the center of the puff pastry to help it deflate—you’re creating a well for your tart toppings. Brush the pastry all over with the remaining tablespoon of olive oil. Scatter the parmesan cheese over the top, leaving a ½-inch border. Scatter the asparagus on top of the parmesan cheese and discard any liquid in the bowl. Create four small wells in the asparagus.
Working one at a time, crack an egg into a small bowl (see PRO TIP below). Gently pour the egg into one of the wells. Repeat with the remaining 3 eggs. Season the eggs with salt. Return the tart to the oven and bake for 10 to 12 minutes, or until the egg whites are set and the yolks are still runny and the puff pastry is golden brown and puffed. The eggs will continue to cook slightly while the tart cools. Remove the tart from the oven.
Dollop some pesto on the asparagus tart and top with ribbons of prosciutto. Finish with more parmesan cheese and some freshly grated lemon zest. Sprinkle with flaky sea salt and more black pepper. Serve right away.