1. Line 2 large cookie sheets with foil. In small bowl, with mixer at high speed, beat egg whites, cream of tartar, and salt until soft peaks form. Beating at high speed, gradually sprinkle in sugar, 2 tablespoons at a time, beating until sugar dissolves. Add vanilla; continue beating until meringue stands in stiff, glossy peaks.
2. Preheat oven to 200°F. Spoon meringue into decorating bag with medium-sized star tube. Pipe meringue into 3" by 3/4 fingers, about 1 inch apart, onto foil-lined cookie sheets.
3. Bake cookies 1 hour or until set. Cool on cookie sheets on wire racks 10 minutes. With small spatula, carefully loosen and remove cookies from foil; cool completely on wire racks.
4. When cookies are cold, in small saucepan over low heat, heat chocolate and shortening, stirring occasionally, until melted and smooth; remove saucepan from heat. Dip one end of each cookie into melted chocolate mixture; let dry on wire rack over sheet of waxed paper. Store cookies in tightly covered container.