Beat 1 cup unsalted butter or margarine, 1 cup white sugar, 6 tablespoons cocoa powder and 1/4 teaspoon ground cinnamon in a large bowl . Beat in egg. Stir 2 teaspoons coffee powder, 1 teaspoon vanilla and 1 teaspoon water in a cup to dissolve coffee. Beat into butter mixture. On low speed, beat in 2 cups flour just until blended.
Divide dough in half and shape into a disk. Wrap and chill until firm.
Preheat oven to 375 degrees F (190 degrees C).
Have ready a 3 inch star cookie cutter. Roll dough on a well floured rolling surface to about 1/8 inch thickness. Cut out stars and place 1 inch apart on an ungreased cookie sheet. Bake for 8 minutes or until crisp.
Put 4 cups confectioners' sugar in a medium sized bowl, gradually stir in hot milk until smooth, stir in 3 tablespoons butter or margarine until blended then add remaining ingredients. Makes about 3 cups. Spoon icing into a corner of a plastic bag, snip off the tip of the corner, drizzle zigzag design on cookies.