Preheat the oven to 350°F (175°C).
Whisk together the flour, baking powder, and salt in a medium bowl until evenly combined.
In a bowl, beat the 5 egg yolks with 3/4 cup granulated sugar on high speed until pale and thick. Stir in 1/3 cup whole milk and 1 teaspoon coconut extract.
Pour the egg yolk mixture into the dry ingredients and gently fold until just combined; do not overmix.
In a clean bowl, beat the 5 egg whites on high speed until soft peaks form, then gradually add the remaining 1/4 cup granulated sugar and continue beating to stiff peaks.
Gently fold the beaten egg whites into the batter in two additions until uniform and no large streaks remain.
Pour the batter into an ungreased 9×13 metal baking dish and smooth the top.
Bake for 25–35 minutes, or until a toothpick inserted in the center comes out clean. Remove from oven and cool completely on a rack.
Make the three-milk mixture by whisking together 13 oz canned unsweetened coconut milk, the 14-ounce can of sweetened condensed milk, 1/4 cup whole milk, and 1/2 teaspoon coconut extract.
When the cake is cool, poke holes all over the surface with a fork and slowly pour the milk mixture evenly over the cake, allowing it to absorb.
Refrigerate the soaked cake for several hours or overnight to allow the milks to soak in.
Whip the heavy whipping cream with the powdered sugar and vanilla extract until stiff peaks form, then spread the whipped cream evenly over the chilled cake.
Sprinkle toasted flaked coconut over the top before serving.