Preheat your oven to 350°F (180°C) and prepare a 13x9-inch baking dish by spraying it with baking spray that contains flour. This will help the cake come out easily once baked.
In the bowl of a stand mixer fitted with a paddle attachment, combine unsalted butter, granulated sugar, and lime zest. Beat on medium speed until the mixture becomes fluffy, about 3 to 4 minutes. Be sure to pause occasionally to scrape the sides of the bowl, ensuring everything is well combined.
Next, add the egg yolks, one at a time, to the butter mixture, beating well after each addition. Follow this by beating in the vanilla extract and coconut extract, allowing the flavors to blend.
In a separate medium bowl, whisk together the cake flour, baking powder, kosher salt, and baking soda. This dry mixture will give your cake structure.
In a small bowl, combine the coconut milk and lime juice. You’ll add this mixture to the cake batter alternately with the flour mixture, starting and ending with the flour mixture. Mix on low speed just until everything is combined, being careful not to overmix.
Transfer the batter to a large bowl. Clean the stand mixer bowl and switch to the whisk attachment. Beat the egg whites on high speed until stiff peaks form. This step is crucial as it helps the cake to rise and become light.
Gently fold the beaten egg whites into the batter in three additions. This folding technique adds air, keeping your batter light and fluffy.
Pour the batter into the prepared baking dish, smoothing the top if necessary. Bake for about 30 minutes or until a wooden pick inserted in the center comes out clean. Let the cake cool completely in the pan on a wire rack.
Once cooled, transfer the cake to a cutting board. With a long serrated knife, level the cake to about 1 inch tall. Cut the cake into two rectangles measuring 8¼x6 inches. Spread the strawberry preserves onto one rectangle, then top it with the other rectangle.
Carefully cut the cake stack into 12 squares, each about 2 inches wide. Place these squares on a wire rack set over a rimmed baking sheet, leaving space between each square.
Freeze the cake squares until they are firm, which should take about 45 minutes to 1 hour. This step ensures that the coating will adhere better.
While waiting, prepare the Strawberry Glaze in a tall, narrow container. Working with one cake square at a time, use a wooden skewer to dip each square into the glaze, letting any excess drip back into the container. Return them to the wire rack and allow them to set for about 30 minutes.
Pour the unsweetened flaked coconut onto a small baking sheet. Roll each cake square in the coconut until fully coated, then place them on your desired serving dish.
Let the lamingtons sit at room temperature for 1 to 2 hours before serving. They can be stored in an airtight container in the refrigerator for up to 3 days.