How to Make It
Preheat oven to 350°F. Combine flour, baking powder, and salt in a large bowl; set aside.
Combine egg yolks and 3/4 cup of the sugar in the bowl of an electric stand mixer. Beat on high speed until fluffy and pale yellow, about 30 seconds. Add coconut extract and 1/3 cup of the coconut milk, beating on medium-low speed until combined, about 30 seconds. Pour egg yolk mixture over flour mixture, and stir until just combined. Clean mixer bowl.
Beat egg whites with mixer fitted with whisk attachment on high speed until soft peaks form, about 1 minute. With mixer running, gradually add 1/4 cup of the sugar, and beat on high until stiff peaks form, about 1 minute. Stir about one-third of the egg white mixture into cake batter; fold in remaining egg white mixture. Spoon batter into a lightly greased 13- x 9-inch baking pan.
Bake in preheated oven until a wooden pick inserted in center comes out clean, about 20 minutes. Cool in pan on a wire rack about 10 minutes.
Meanwhile, combine sweetened condensed milk, 1/4 cup of the heavy cream, and remaining coconut milk in a medium bowl. Pierce top of warm cake several times with a fork; gradually pour milk mixture over warm cake, about 1/2 cup at a time. Allow mixture to soak into cake completely before adding additional milk mixture. Cover and chill 4 hours.
Beat coconut liqueur, remaining 2 cups heavy cream, and remaining 3 tablespoons sugar with an electric mixer fitted with whisk attachment on medium-high speed until stiff peaks form, 2 to 3 minutes. Spread mixture over cake, and garnish with toasted shredded coconut and lime peel strips.