In a mixing bowl, combine the Parmesan, 1 whole egg, 1 egg yolk, lemon zest and dill. Season well with salt and pepper, then mix until combined. Wash the asparagus and trim off the woody ends (around 2.5cm/1 inch from the bottom of each spear).
Unroll the puff pastry and cut it into 4 equal rectangles. Place them on the prepared baking tray.
Using a sharp knife, score a 1cm border around the edge of each rectangle, being careful not to cut all the way through the pastry.
Divide the cheese mixture evenly between the tarts, spreading it within the scored border. Arrange the asparagus spears over the cheese mixture.
Beat the remaining egg and brush it around the pastry borders.
Bake for 20 minutes, or until the pastry is golden, crisp and puffed up around the edges