Preheat oven to 375°F (190°C). Line baking sheets with parchment paper.

In a large bowl, beat butter and confectioners’ sugar with a mixer at medium speed until fluffy, 2 to 3 minutes, stopping to scrape sides of bowl. Add egg, rum, and vanilla, beating until combined.

In a medium bowl, whisk together flour, baking powder, salt, cinnamon, and nutmeg. With mixer on low speed, gradually add flour mixture to butter mixture, beating until a dough forms.

Divide dough in half; cover one half with plastic wrap. On a lightly floured surface, roll uncovered dough to ¼-inch thickness. Using 5 graduated-size, star-shaped cookie cutters*, cut out 3 cookies for each size, rerolling scraps as needed. Repeat with remaining dough. (Each tree will have 15 cookies.) Working in batches and using a small offset spatula, place cookies, sorted by size, at least ½ inch apart on prepared pans.

Bake until edges are just beginning to brown, 7 to 10 minutes. Let cool on pans for 1 minute. Remove from pans, and let cool completely on wire racks.

Transfer about 1 cup (206 grams) Spiked Royal Icing to a small bowl; add 2 tablespoons (30 grams) water, stirring until icing is flood-consistency. (If icing is too thick, add more water, and if it’s too thin, add more confectioners’ sugar. For reference, flood-consistency icing has the appearance of runny honey.) Dip tops of cooled cookies into flood-consistency Spiked Royal Icing, letting excess drip off. Let stand until dry, at least 4 hours. (Don’t rush this process—if you start to move your cookies or ice with new details, you’ll risk smudging.) Spoon 1 cup (206 grams) Spiked Royal Icing into a small piping bag fitted with a small round piping tip (Wilton No. 2) or cut a ⅛-inch opening. Pipe desired decorations onto dipped cookies. (See Decorating Techniques on page 41.) Refill piping bag as needed. Garnish with sprinkles, if desired. Let stand until dry, at least 1 hour.

For first set of 15 cookies, start with largest star and stack cookies in order of size, spiraling them so points do not line up; use Spiked Royal Icing as an adhesive, piping a little icing onto the center point of each cookie before layering with the next. Stand final cookie upright as a tree topper, securing with Spiked Royal Icing. (To help top cookie stand up, gently grate bottom points until flat and dot with icing before securing to the top of the tree.) Repeat procedure with second set of 15 cookies.