Preheat the oven to 350°F degrees. Racks in the top and bottom third.
Sift the flour, baking powder, salt, cloves cinnamon, and allspice into a medium bowl - set aside. In the bowl of an electric mixer (or by hand), cream together the butter and sugar. Add the egg and vanilla. Blend well, scraping down the side of the bowl a few times along the way. Add the currants and walnuts. Add the dry ingredients in three batches, alternating with the milk. Chill (covered) for one hour or longer, up to a day.
Drop the cookie dough (one level tablespoon at a time) onto ungreased cookie sheets, leaving an inch or so between cookies. Dampen your fingers with a bit of water and gently flatten the dough. Bake for 12 -15 minutes or until the bottoms of the cookies are deeply golden. Cool on a wire rack.
While the cookies are cooling, make the icing, and consider making double if you like a thick layer. Whisk the powdered sugar, heavy cream, and vanilla together in a small bowl - use immediately. Frost each cookie with an off-set spatula (or pastry bag) - but not until they are completely cool.