Beat shortening and brown sugar in a large bowl at medium speed with an electric mixer 5 minutes or until fluffy; add egg, beating until blended.
Stir together flour and next 3 ingredients in a bowl; add to shortening mixture alternately with sour cream, beginning and ending with flour mixture. Add stuffed dates, stirring gently until coated. Place dates, 1 at a time, on lightly greased baking sheets.
Bake dates at 375° for 10 to 13 minutes; remove to wire racks to cool. Spread Browned Butter Frosting evenly over each date.