Preheat oven to 400°F (200°C). Line 3 baking sheets with parchment paper.

In a light-colored medium saucepan, melt butter over medium heat. Cook, stirring constantly, until butter turns a medium-brown color and has a nutty aroma, 5 to 8 minutes. Remove from heat, and pour browned butter into the bowl of a stand mixer fitted with the paddle attachment. Let cool until butter starts to solidify, about 20 minutes.

Add confectioners’ sugar to browned butter, and beat at medium speed until combined, about 1 minute, stopping to scrape sides of bowl. Add egg and vanilla, beating until combined.

In a medium bowl, whisk together flour, baking powder, cardamom, and salt. 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 heavily floured surface, roll half of dough to ¼-inch thickness. (Lightly flour top of dough if it sticks to rolling pin; keep remaining dough covered with plastic wrap.) Using desired holiday cutters, cut dough, rerolling scraps as necessary. Using a small of set spatula, place at least 1 inch apart on prepared pans.

Bake, one batch at a time, until bottoms are lightly golden, 7 to 8 minutes. Let cool on pan for 5 minutes. Remove from pan, and let cool completely on wire racks. Decorate cooled cookies as desired with Royal Icing.