Beat 3/4 cup butter in a mixing bowl with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, ginger, baking soda, cloves, nutmeg, and allspice; beat until combined. Beat in molasses, water, and rum or milk. Beat in as much flour as you can with the mixer. Stir in remaining flour with a wooden spoon. Divide dough in half; cover and chill at least 3 hours or until easy to handle
Roll each portion of dough on a lightly floured surface until 1/4 inch thick. Cut with a floured 4-inch scalloped or plain cookie cutter. Place cutouts 1 inch apart on a greased cookie sheet. Bake in a 375 degree F oven for 9 to 11 minutes or until edges are firm and bottoms are just lightly browned. Cool on cookie sheet for 1 minute. Transfer to wire racks to cool completely.
Meanwhile, for the Lemon Frosting, combine powdered sugar, 1 tablespoon softened butter or margarine, and finely shredded lemon peel in a small mixing bowl. Stir in lemon juice. Stir in 2 to 3 teaspoons milk until frosting is smooth and of drizzling consistency. Drizzle cookies with Lemon Frosting. Makes about 24.