For cookies: Whisk first 6 ingredients in medium bowl. Beat butter and sugar in large bowl until fluffy. Beat in molasses. Beat in dry ingredients. Gather dough; divide into 4 pieces. Shape into disks. Wrap; chill at least 2 hours and up to 2 days.
Preheat oven to 350°F. Line 2 baking sheets with parchment. Roll out 1 dough disk to 1/8-inch thickness. Using 3 1/2-inch cutter, cut out cookies. Transfer to sheet. Gather scraps; chill.
Bake cookies until almost firm in center, 12 minutes. Cool on sheets 2 minutes, then cool on racks. Repeat, using all dough. Do ahead Store airtight at room temperature up to 3 days or freeze up to 2 weeks.
For glaze and decoration: Bring first 2 ingredients to simmer. Cover; chill 5 hours. Strain. Place powdered sugar in bowl. Whisk in half and half by spoonfuls until glaze is spreadable. Frost cookies; decorate. Let stand until glaze sets. Do ahead Store between sheets of waxed paper in airtight container at room temperature up to 3 days.