For the Cake:

1. PREP: Position a rack in the center of the oven and pre-heat the oven to 350 degrees. Spray the bottom and sides of a 9×13-inch baking dish with a baking spray (or use butter) and then line the bottom of the baking dish with parchment. Spray (or butter) the parchment paper.
2. DRY INGREDIENTS: Sift the flour, baking powder, and salt into a medium bowl and set aside.
3. BEAT EGG YOLKS & SUGAR: Next, separate the eggs: put the whites in a medium bowl and the yolks in a large bowl. Using an electric mixer, beat the yolks with 3/4 cup of the sugar on medium speed until the mixture is pale and creamy, about 2 minutes.
4. ADD MILK & VANILLA: Add the milk and vanilla and beat until combined, 1 minute more.
5. BEAT EGG WHITES + REMAINING SUGAR: Next, using clean beaters, beat the egg whites, gradually increasing the speed to high, until they reach soft peaks, 2 to 3 minutes. Slowly add the remaining 1/4 cup sugar and continue to beat on high, until you reach firm but not glossy, dry peaks, about 1 to 2 minutes more.
6. WHISK IN DRY INGREDIENTS: Then whisk in a third of the dry ingredients into the yolk mixture until thoroughly combined. Next, using a rubber spatula, gently fold in a third of the egg whites. Fold in the remaining dry ingredients and egg whites, alternately, in two more batches each, until fully incorporated.
7. BAKE: Pour the batter into the prepared dish or pan and bake until a toothpick inserted in the center comes out clean, 20 to 25 minutes. The cake will be very springy.
8. COOL: Let the cake cool in the pan on a rack for 10 minutes, and then invert it onto the cooling rack.  Remove the parchment paper and let cool completely. When the cake is cool, return it to the baking dish or pan (my preferred method), or invert it onto a rimmed dish or platter.
9. MAKE SYRUP: While the cake is cooling, make the soaking syrup. In a medium saucepan, stir together the condensed milk, whole milk, coconut milk, and salt until well blended. Cook over medium-low heat, gently stirring to avoid burning, until bubbles begin to form around the edges, about 3-5 minutes. Don’t boil it. Remove from the heat and pour the mixture into a heatproof 4-cup measuring cup to cool slightly.
10. ADD TO CAKE: Using a toothpick or a skewer, prick the cake, going all the way through the bottom, in approximately 1/2-inch intervals. Pour the soaking liquid slowly over the cake, pausing to let it soak in before adding more. You might not need all of the syrup.
11. SOAK: Cover loosely and refrigerate until the cake is well chilled, at least 2 hours and up to 24 hours.
12. ASSEMBLE: When ready to serve, use an electric mixer to whip the cream, 2 tablespoons of powdered sugar, the rum, and the vanilla extract together until soft peaks form. Stir in the lime zest (reserving some to sprinkle on top. Adjust for taste and then spread the whipped cream over the cake. Next, sprinkle the toasted coconut over top.
13. SERVE & STORAGE: You can serve immediately or keep it covered in the refrigerator for 2-3 days.