Whisk flour, cornstarch, spices, and salt in a bowl.
Put butter and 1/3 cup sugar into the bowl of an electric mixer fitted with the paddle attachment.
Mix on medium speed until pale and fluffy.
Mix in rum and vanilla.
Reduce speed to low, and gradually mix in flour mixture.
Divide dough in half.
Place each on a piece of parchment paper; shape dough into logs.
Fold parchment over dough; using a ruler, roll and press into a 1 1/4-inch log.
Wrap in parchment.
Chill in freezer 30 minutes (up to 1 month).
Preheat oven to 350 degrees.
Cut into 1/4-inch-thick rounds; space 1 inch apart on parchment-lined baking sheets.
Bake until just golden, about 15 minutes.
Transfer cookies to wire racks; let cool 10 minutes.
Gently toss warm cookies with remaining 2/3 cup sugar in a resealable plastic bag.
Cookies can be refrigerated in airtight containers up to 2 weeks.