Add the almond extract. In another bowl, sift together the flour, baking soda, and salt. With a wooden spoon, gradually stir the flour mixture into the shortening. The dough should be fairly firm. Wrap in waxed paper and refrigerate for 4 hours.
Place cookies on an ungreased cookie sheet.
Top each cookie with an almond half. Brush cookies lightly with beaten egg. Bake for about 10 minutes, or until light golden brown. Add the sweet potatoes, and deep-fry until brown. Add the sugar and cook, stirring, until it dissolves.