In a mixer bowl, beat together butter, confectioners' and brown sugars on medium speed until light, about 2 minutes. Beat in extract, then flour and salt until a dough forms. Wrap well in plastic wrap, flatten into a disk and chill until firm, about 1 hour or up to 1 week.

Heat oven to 325F and place rack in center. Line two baking sheets with parchment paper. On a floured work surface, using a floured rolling pin, roll out dough to about 1/8 inch thick, occasionally running a spatula under dough to keep from sticking to work surface. Using 1-1/4-, 1-3/4-, 2-1/2- and 3-inch star cookie cutters, cut out ten 1-1/4-inch and 20 of each of the other sizes; transfer to prepared sheets. Bake, one sheet at a time, until just golden around edges, 8 to 10 minutes. Transfer sheets to a wire rack and let cool completely.

Transfer icing to a decorating bag fitted with a small plain. Secure end of bag with a rubber band, squeezing out air. (When not using, place, tip side down, in a tall glass with a damp paper towel in bottom so icing doesn't harden in tip.)

Working with one cookie at a time, pipe icing around rim of each star and sprinkle with sanding sugar. Let stand until set, about 30 minutes.

For each tree, using a dab of icing to adhere, stack 2 large, 2 medium, 2 small and 1 tiny star; top with a gumdrop. (Can be made ahead. Store in an airtight container at room temperature up to 2 days.)