Unfold puff pastry on a piece of parchment paper. Roll it in one direction so it's a rectangle approximately 14" by 10" and the lines from it being folded are gone.
Using a small, sharp knife, score the puff pastry approximately 1/2 inch from the edge of the puff pastry, without cutting through the dough to the bottom. This will allow the edge to rise and create a puffy, crispy crust.
Spread the shredded mozzarella (1 cup) on top of the puff pastry, trying to stay inside the scored lines.
Top with the grated parmesan cheese (1/4 cup).
Arrange the asparagus spears on top, close together, alternating the directions they're facing so the ends and tips alternate.
Drizzle the top with the olive oil and sprinkle with a little bit of kosher salt and black pepper.
Bake at 400 for approximately 20 minutes, or until edges of pastry are golden brown.
Allow to cool for at least 5 minutes and slice into quarters. Serve immediately.
Notes
To trim asparagus: hold it in the middle and grab the tough end. Bend, and snap! (anyone catch that reference?) It will snap at the place where it becomes less tough. Then, if you want, you can arrange them and trim them to a uniform size.
Ingredient Substitutions: Other cheeses, such as gruyere, goat cheese, or fontina, can be used. Other quick-cooking veggies, such as zucchini, tomatoes, summer squash, small broccoli florets, or mushrooms can be used instead of asparagus.
The provided nutrition information does not include any added sodium from seasoning to taste, any optional ingredients, and it does not take brands into account. Feel free to calculate it yourself using this calculator or by adding the recipe to Yummly.