Heat oven to 350°F (175°C). Line two baking sheets with parchment. Cream the butter and sugar until pale and fluffy, about 3 minutes, then beat in the egg and vanilla.

Sift together flour, cornstarch, baking powder, and salt. Mix into the butter mixture on low just until combined. Shape into a disc, wrap, and chill at least 1 hour.

Roll the dough between parchment to ¼ inch thick. Cut tree shapes and place on the prepared sheets.

Bake 8 to 10 minutes, until edges are just barely golden. Cool on the sheet 5 minutes, then transfer to a rack to cool completely.

Whisk powdered sugar, milk, vanilla, and green food coloring into a thick, pipeable icing. Pipe onto the cookies and add sprinkles while wet. Let icing set about 15 minutes.