Step 1

For the cookies: Stir together the flour, baking powder and salt in a medium bowl. Beat the butter and granulated sugar in a large bowl with an electric mixer on medium-high speed until light and fluffy, 2 to 3 minutes. Scrape down the sides of the bowl with a rubber spatula, then add the egg and vanilla and beat until combined, about 2 minutes. Add the flour mixture and beat on low speed just until the dough comes together in a smooth mass, 1 to 2 minutes. Wrap the dough in plastic wrap and flatten into a disk. Refrigerate until firm but still malleable, about 1 hour.

Step 2

Preheat the oven to 350 degrees. Line 3 baking sheets with parchment.

Step 3

Cut the dough in half. Roll one half on a lightly floured surface to between 1/8 and 1/4 inch thick. Use a 2 1/2-inch cutter to cut as many rounds as you can. Transfer to the baking sheets, spacing about 1 inch apart. Repeat with the other piece of dough. Gather the scraps together and roll once more to get additional rounds, if desired. You should get about 30 to 34 cookies. (If the dough softens too much while rolling or rerolling, chill it for 15 minutes before proceeding.)

Step 4

Bake 2 pans of the cookies, rotating the sheets halfway through, until golden on the bottoms and edges, 10 to 12 minutes. Transfer the sheets to wire racks and let the cookies cool completely. (They will crisp as they cool.). Repeat with the last pan of cookies.

Step 5

While the cookies cool, make the shark fins. Cut the chocolate-peppermint candies in half with a paring knife. Use a small round cutter to cut a wedge from the right side of each candy to make the inner curve of a shark fin (see Cook's Note). Set aside.

Step 6

For the icing: Beat the confectioners' sugar, egg whites, milk, vanilla and salt in a large bowl with an electric mixer on medium-high speed until thick and glossy, about 2 minutes, adding 1 to 2 tablespoons more milk as needed until the icing thickly coats a spoon and drops down in large drizzles.

Step 7

Transfer a little of the icing to each of 3 small bowls (3 to 4 tablespoons per bowl). Tint the remaining icing with 3 or 4 drops of blue food coloring. Leave one small bowl white, tint the second bowl teal and the final one green.

Step 8

Spread the blue icing about 1/2 inch thick on a rimmed baking sheet. Working quickly so the icing doesn’t harden, drizzle a little more than half of the icing from each of the small bowls over the top. Place as many cookies as will fit (about half) face down into the icing. Give the cookies a quarter turn before removing to facilitate swirling. Let the excess icing drip back onto the sheet. Place the cookies icing-side up on racks as you finish them. Drizzle the remaining small bowls of icing over the blue again and repeat with remaining plain cookies.

Step 9

Sprinkle the cookies while the icing is still wet with blue sanding sugar and pearl sugar, then stick a shark fin in the middle of each. Let the icing set before serving, at least 1 hour.