Step 1Bring a large pot of water to a simmer. Trim the asparagus and cook for 2 minutes. Shock in ice water and drain, then pat dry.
Step 2Heat the oven to 400 degrees. Cut the puff pastry into 4 (6- by 4-inch) rectangles to make tart shells and place on a baking sheet. Heat the oven to 400 degrees.
Step 3Combine the mascarpone with the grated Parmesan. Spread the mixture evenly over the tarts, leaving a three-quarters-inch border.
Step 4Divide the asparagus among the tarts on top of mascarpone, placing 4 to 6 on each. Drizzle each tart with 2 teaspoons olive oil.
Step 5Bake 20 to 25 minutes, until crisp and golden. Sprinkle each tart with 1 tablespoon shaved Parmesan; serve.