Blanch the asparagus: fill a large bowl with ice water. Bring a pot of water to a boil. Add the asparagus to the boiling water and cook 2-5 minutes until bright green and crisp-tender. Drain and transfer the asparagus to the ice water to stop the cooking. Drain and pat dry.
Assemble the tart: Preheat the oven to 400F. Roll out the puff pastry sheet to a 10 by 16-inch rectangle on a baking mat or large piece of parchment. Transfer the baking mat/parchment with rolled puff pastry to a sheet pan.
Spread dijon mustard in an even layer on the puff pastry, leaving a one-inch border around the sides.
Toss together the mozzarella and parmesan cheeses and sprinkle evenly over the layer of the dijon mixture. Lay the asparagus side-by-side across the cheese. Sprinkle with salt and pepper and bake 15-20 minutes until pastry is golden brown and cheese is puffy.