Cream shortening and white and brown sugar. until fluffy. Add egg, milk, and vanilla, beating well. Sift dry ingredients together and add to creamed mixture. Mix well. Chill dough at least an hour.
Meanwhile, make date filling by combining chopped dates, sugar and water in a saucepan. Bring to a boil. Cover and simmer about five minutes, stirring occasionally. Add lemon juice and salt. Cool.
On floured surface (or pastry cloth), roll half of the dough at a time to a little less than 1/8 inch. Cut with 2 1/2 inch round cookie cutter. (Recipe recommends cutting a small hole in the center of half the cookies for the top with a thimble. My mom has always skipped this step.) Place 1 tablespoon of prepared date filling in the center of each plain cookie bottom. Top with another round cookie (with small hole if you are following original recipe). Press the edges of each filled cookie together with the tip of a fork to seal. Bake on an ungreased cookie sheet at 375° for about 10 to 12 minutes.