Place asparagus pieces (except the tips) in a microwave-safe container. Add a tablespoon or so of water and microwave on high for 3 minutes to steam the asparagus. Drain and set aside. Preheat oven to 400F.

Heat a skillet on medium-high heat and add the mushrooms. Cook until the mushrooms begin to release liquid, then add the shallot and stir. Add the cooked asparagus. Cook until the shallots are soft and separated, the mushrooms are cooked and beginning to brown, and the liquid is gone.

Turn off the heat and immediately add the mascarpone cheese and thyme to the pan. Stir until the cheese is melted and the vegetables are coated with cheese.

On a lightly floured surface, roll out the puff pastry to about a 10x14-inch rectangle. Transfer to a baking sheet and fold the edges to make a slight crust. Lightly prick the surface with a fork, then sprinkle with ¼ cup of Parmesan cheese.

Spread the cooked vegetable mixture evenly over the crust. Add the uncooked asparagus tips and dot with fresh mozzarella cheese. Sprinkle the remaining Parmesan cheese evenly over everything.

Bake 20 minutes.

Remove from oven and cool slightly, then cut into squares and serve warm.