Trim the ends off of one large leek and slice it in half lengthwise. Rinse the leek under cool running water to remove any grit that is caught between the layers of the leek. Slice the leek into thin half-moon slices and set aside.
Wash and trim the woody ends off of one bunch of asparagus. Set aside.
In a small bowl, mix together the mayonnaise, mustard and one tablespoon of minced tarragon. Set aside.
On a lightly floured surface, gently roll out one sheet of puff pastry into a 9"x12" rectangle. Brush the pastry with the mustard mixture, leaving about a ½ inch edge around the border of the pastry. Top the tart with the asparagus, then the leeks.
In a separate small bowl, whisk together the egg and cold water. Brush the edges of the pastry with the egg wash. Bake the tart at 400°F for 25 minutes. As soon as the tart comes out of the oven, top it with the shredded parmesan cheese and the remaining one tablespoon of minced tarragon.
Let the tart rest for 5 minutes before slicing and serving. Enjoy!
