Combine 1 cup butter and sugar in large bowl; beat at medium speed until creamy. Add egg, whipping cream, 1 teaspoon rum extract and vanilla; continue beating until well mixed. Reduce speed to low; add all remaining cookie ingredients. Beat until well mixed.
Divide dough in half. Shape each half into a ball; flatten slightly. Wrap each in plastic food wrap; refrigerate until firm (2 to 3 hours).
Heat oven to 375 degrees F. Roll out dough on lightly floured surface, one-half at a time (keeping remaining dough refrigerated), to 1/4-inch thickness. Cut with 2 1/4-inch snowflake-shaped cookie cutters. Place 1 inch apart onto ungreased cookie sheets. Bake for 6 to 8 minutes or until edges are lightly browned. Remove to wire cooling rack. Cool completely.
Combine powdered sugar, 2 teaspoons butter, corn syrup and 1/4 teaspoon rum extract in small bowl. Beat at medium speed, adding enough water for desired glazing consistency.
Place wire cooling rack with cookies over waxed paper. Brush glaze over cookies; decorate as desired. Let stand until set.