Use of duck fat at the time of cooking?
I think the usual use for drippings is incorporating the fat into a gravy or pan sauce.  A bit of flour, a bit of water, salt and whatever seasonings.  I don't recall if it is common for duck specifically, but I see no reason it an't work.

As for your pastry idea, I don't think it usually works that way, the physical disturbance of the dripping and the uneven distribution might make the pastry texture not come out quite so well. (I assume there's a reason pastry isn't commonly used to catch drippings).  

If you still want to try something similar, you might be interested in yorkshire pudding - which was historically used to catch and use the drippings from a roast or similar, and being intended for this purpose it will probably work better than a random pastry recipe.