In a large bowl, whisk together flour, baking powder, and salt.
Using an electric mixer, beat butter and sugar until light and fluffy, about 3 minutes. Beat in egg, then vanilla.
Reduce mixer speed to low and gradually add flour mixture, mixing just until incorporated. Shape dough into 4 disks and roll each between 2 sheets of wax paper to 1/8-inch thick. Chill until firm, 30 minutes in the refrigerator or 15 minutes in the freezer.
Heat oven to 350°F. Line baking sheets with parchment paper. Using floured cookie cutters, cut out cookies. Place on prepared sheets. Reroll, chill, and cut the scraps.
Bake, rotating pans halfway through, until cookies are lightly golden brown around the edges, 10 to 12 minutes. Let cool on the sheets for 5 minutes before transferring to wire racks to cool completely.
Outline and fill each cookie (leaving the polar bear’s sweater area unfilled) with white royal icing. To make polar bear noses, pipe a mound of stiff royal icing in one spot and let sit until almost dry, then press on mini brown M&M’s. Pipe on faces, then decorate sweaters, scarf, and hat as desired using tinted royal icing.