In saucepan, combine dates, the first 1/2 cup granulated sugar, the water, and the 1/4 teaspoon salt.
Bring to boiling; reduce heat, cover, and simmer about 5 minutes; stir frequently.
Add lemon juice; cool.
In mixer bowl, cream vegetable shortening, the remaining granulated sugar, and the brown sugar.
Add egg, milk, and vanilla; beat well.
Combine remaining ingredients; add to creamed mixture and mix well.
Chill at least 1 hour.
On lightly floured surface, roll out half the dough to 1/8 inch thickness; cut with 2 1/2 inch round cutter.
Transfer half the rounds to ungreased cookie sheet.
Top each with 2 teaspoons date mixture.
Top with remaining cookie rounds.
Press edges together with tip of teaspoon to seal.
Repeat with remaining half of dough.
Bake at 375° for 10 to 12 minutes.