Preparation

Make the ice cream

• Put an 8×8- or 9×9-inch pan in the freezer. If the ice cream was frozen after churning (or if you’re using store-bought ice cream), let it sit at room temperature for about 10 minutes to soften, then put it in the bowl of a stand mixer fitted with the paddle attachment. Add the coconut and mix on low speed until combined. (Alternatively, combine the ice cream and coconut in a large bowl using a silicone spatula.) Spread the ice cream in the frozen pan, cover with plastic wrap directly on the surface of the ice cream, and freeze until scoopable, about 4 hours or overnight.

Make the cookies and assemble the sandwiches

• Position a rack in the center of the oven, and heat the oven to 350°F. Line two large rimmed baking sheets with parchment; set aside. Melt the butter and chocolate in a double boiler over gently simmering water. Stir to combine. Let cool to room temperature.
• Sift the flour and baking powder onto a sheet of parchment. Add the salt.
• In the bowl of a stand mixer fitted with the whisk attachment, whip the eggs, sugar, and vanilla until thick. On low speed, mix in the cooled chocolate mixture. Add the flour mixture, and stir until just combined. Let the batter rest for 5 minutes.
• Scoop heaping tablespoons of dough (about 1-1/4 inches in diameter), and place them at least 2 inches apart on the prepared baking sheets. Bake for 5 minutes. If the cookies are still domed and not flattened, press gently with the bottom of a measuring cup or with your fingers. Bake about 5 minutes more. When done, they will look set and no longer shiny on top. Let cool, then lift off the baking sheets with a spatula.
• Put the toasted coconut in a small bowl. Place 10 of the cookies, bottom side up, on a baking sheet. Place a large scoop of ice cream on top of each cookie. Top with a second cookie and gently press down on the top cookie to adhere the cookies together. Roll the edges of the sandwiches in the coconut. Freeze until ready to serve.