In large bowl, beat granulated sugar and 1 cup butter with electric mixer on medium speed until light and fluffy. Beat in molasses and egg until well blended. On low speed, beat in remaining cookie ingredients until well mixed. If necessary, cover with plastic wrap; refrigerate 1 hour for easier handling.
Heat oven to 350°F. On lightly floured surface, roll out 1/4 of dough at a time to 1/8-inch thickness. (Keep remaining dough refrigerated.) Cut with floured 2 1/2-inch round or desired shape cookie cutters. On ungreased cookie sheets, place 1 inch apart.
Bake 6 to 9 minutes or until set. Cool 1 minute; remove from cookie sheets to cooling racks. Cool completely, about 15 minutes.
In small bowl, beat 1/4 cup butter and the shortening on medium speed until light and fluffy. Beat in vanilla and dash salt. On low speed, gradually beat in powdered sugar until smooth. On high speed, beat in 1 tablespoon milk until light and fluffy. Add enough additional milk until frosting can be spread or piped. Frost or decorate cookies as desired. Let stand until frosting is set, about 30 minutes. Store in loosely covered container.