Make the pastry for these from the recipe on this page. You do not need to pre-bake the shells.
Combine the egg, brown sugar, butter, vanilla extract and pecans in a medium bowl until well mixed. Fill each dough shell no more than 2/3 full. (Be sure NOT to overfill, as the filling will bubble up over the edges and make a fine mess. You will probably have some filling left over, but you can't cut down on the amounts for a single batch because of the egg. If you're making multiple batches of these tarts then make one less multiple of the filling than you do of the crust.) Bake for 25 minutes or until the tops are golden. Transfer the pans to a wire rack to cool for at least 10 minutes, then run a sharp knife around the outside edges of the tassies to release them from the pans. If you do have an overflow, be sure to run the knife around the edges right away when they come out of the oven but then wait to actually remove them until they've cooled.