Heat oven to 350°F. Grease with shortening and flour large cookie sheet. In large bowl, break up cookie dough. Sprinkle vanilla and rum extract over dough; mix well. Stir toasted almonds into dough.
Divide dough into 4 equal portions; shape each into 8x1-inch log. On cookie sheet, place logs 3 inches apart; flatten each log until about 1 1/2 inches wide.
Bake 15 to 20 minutes or until golden brown. Cool 15 minutes. Reduce oven temperature to 200°F.
With serrated knife, carefully cut each log into 10 (3/4-inch) slices. On same cookie sheet, place slices, cut side down.
Return to oven; bake 1 hour. Remove cookies from cookie sheet to cooling rack. Cool completely, about 20 minutes. Meanwhile, in small microwavable bowl, place chocolate chips. Microwave on High 1 minute. Stir; microwave 1 minute longer, stirring every 15 seconds.
Line cookie sheet with waxed paper. Dip 1/4 of each cookie into melted chocolate; place on cookie sheet. Refrigerate until chocolate is set, about 10 minutes.