Place oven rack in center position and heat oven to 350 degrees F. Grease cookie sheets or line with parchment paper.

In the bowl of electric mixer or by hand, cream together the sugar and shortening. Add the egg and vanilla extract and beat until well combined. In a large bowl, whisk together the flour, baking powder, and salt. Add dry ingredients alternately with the milk to the shortening mixture, beginning and ending with the flour mixture, making sure all of the ingredients are well incorporated. The dough will be soft. Divide the dough into four even pieces, wrap each in plastic wrap, and refrigerate until ready to use.

To make the filling, either process the nuts through a food processor until finely chopped (just a little larger than all-out ground), or chop by hand. The smaller the pieces, the better. Combine the chopped nuts with the sugar and then add in the melted butter. Mix well, making sure there are no large clumps. Add the egg whites and again, mix well.

On a well-floured surface, roll out a piece of dough into a rectangle measuring about 6-inches by 18-inches. Spread 1/4 of the nut filling onto the dough, leaving a small border around the perimeter of the dough. Roll up as you would a jelly roll, with the short ends to the left and right of you, and seal the ends. Cut the roll into 1-inch pieces and place on the cookie sheet. Bake until the tops are just starting to turn light brown, about 15 to 20 minutes. Set on wire racks to cool completely. Repeat with the remaining pieces of dough and filling.

Once the cookies are cooled, prepare the icing by mixing together the powdered sugar, vanilla, and enough milk to achieve the desired consistency. You'll want the icing to be thick enough to not be runny, but still easily spreadable. Ice the tops of the cookies and let set completely before storing in an airtight container.

Makes about 7 dozen cookies.