Cream butter in a large mixing bowl; gradually add 1/2 cup sugar, beating until light and fluffy. Add egg yolk; beat well. Gradually add flour to creamed mixture alternately with milk, beginning and ending with flour, beating well after each addition. Chill 1 hour.
Divide dough into 3 equal portions. Roll each portion to 1/8- inch thickness between two sheets of waxed paper. Cut into 2-inch squares. Place 1 date in center of each square; fold and press dough around date.
Combine pecans and remaining sugar in a small mixing bowl; stir well. Dip cookies in egg whites (at room temperature); roll in pecan mixture.
Place 2 inches apart on greased cookie sheets. Bake at 350° for 20 minutes or until lightly browned. Remove from cookie sheets, and cool on wire racks. Store in airtight containers.