Coconut Sheet Cookie for Sandwich

To make the coconut cookie: preheat the oven to 300°F (150°C). Spray two quarter sheet pans (6.5in. × 4.5in. × 0.5in.) with non-stick cooking spray, line with parchment paper, and spray again. Put the eggs, lime zest, sugar, and salt into the bowl of an electric mixer fitted with the whisk attachment. Whisk on medium speed until thick, pale yellow, and doubled in volume, 8 minutes. Stir in the vanilla extract and gently fold in the coconut. 

Spread the batter evenly into the prepared baking sheets and bake until fragrant and golden brown, 15–18 minutes. Remove from the oven and cool in the pan completely until the cookie is hard when touched, then invert into a large cutting board. 

Transfer one of the sheet cookies onto the quarter sheet pan, inverting it upside down so the browned crust side is at the bottom. Reserve the other sheet until ice cream is ready to be assembled.

Coconut Ice-cream

Put the milk, coconut milk, dried coconut, sugar and salt in a medium saucepan and set over medium heat. Bring to a steady simmer and cook, stirring occasionally, for 8 minutes. Let mixture sit until it comes to room temperature, about an hour. Break yolks with a whisk and set aside until later. 

Bring mixture to boil again and pour ½ cup of the warm milk mixture onto the yolks in a slow, steady stream, whisking constantly. Transfer the yolk mixture back to the saucepan, set over low heat, and cook, stirring constantly, until the mixture is thick enough to coat the back of a spoon and registers 165˚F (75˚C), about 5 minutes. Remove from the heat and strain the mixture through a fine-mesh sieve into a large mixing bowl, then pour the cream through the sieve, squeezing the coconut flakes to extract as much liquid as possible. Stir in the rum. Set the mixture over a larger bowl of ice and water and cool, stirring occasionally, until cool to the touch, about 40˚F (4˚C). Alternatively, cover and refrigerate until cold. 

Transfer the mixture to your ice cream maker and follow the manufacturer’s instructions.

Assembly 

Use an offset metal spatula to spread churned ice cream over the sheet cookie until covered. Place the other sheet cookie on top and gently press down to stick to the ice cream. Cover the entire ice cream sandwich tightly with plastic wrap. Once frozen, remove from freezer and invert onto a large cutting board. Cut ice cream sandwich into square pieces of about 2” x 2”. Cut each square diagonally into two triangles and serve immediately.