Preheat the oven to 350°F (175°C) and line two baking sheets with parchment paper.
Make the dough: Whisk the all-purpose flour, baking powder, and salt together in a medium bowl.
In a large bowl, use an electric mixer on medium speed to cream 1 cup granulated sugar and 1 cup vegetable shortening until light and fluffy. Beat in 1 egg and 1 teaspoon vanilla until combined.
Add the flour mixture and 1 cup milk alternately to the creamed mixture, beginning and ending with the flour mixture, mixing until just incorporated; the dough will be soft.
Divide the dough into four equal portions, wrap each in plastic wrap, and refrigerate for at least 1 hour until firm.
Make the filling: In a large bowl, combine the very finely chopped walnuts and 1½ cups granulated sugar. Stir in the melted butter, then mix in the lightly beaten egg whites until evenly combined.
Assemble: On a well-floured surface, roll one chilled dough portion into a rectangle about 6 x 18 inches. Spread one-quarter of the walnut filling evenly over the dough, leaving a small border.
Tightly roll the dough like a jelly roll from a short side, seal the ends, and cut the roll into 1-inch slices. Place each cookie cut-side up on the prepared baking sheets.
Repeat rolling, filling, and slicing with remaining dough and filling. Bake the cookies until light golden brown, about 30 minutes.
Cool the cookies on a wire rack completely before icing.
Make the icing: Whisk the powdered sugar, 1 teaspoon vanilla, and 4 tablespoons milk in a small bowl until smooth. Dip the tops of the cooled cookies into the icing, allow excess to drip off, and let the icing set completely.