Twist off the crown of pineapple leaves and discard. Slice off the top and bottom ends of the pineapple to make flat surfaces. With a cut side down, use a large chef's knife to remove strips of the peel, following the curve of the pineapple as much as possible. When the peel is removed, cut the pineapple in half lengthwise, then in half again to make quarters. Place a quarter cut side down and cut off the pale-yellow core. Slice the quarter lengthwise into into 3/4 inch pieces, then cut the pieces crosswise into 1/2 inch chunks. Repeat with the the remaining quarters.
Cook the pineapple topping:
In an 8" oven-proof skillet, combine the pineapple chunks, unrefined sugar and grated ginger. Cook over a medium flame, stirring occasionally, until the pineapple chunks are translucent and brown from the sugar and the sauce bubbles thickly, about 15 minutes. You should have around 1 1/2 cups of fruit.
Place the pineapple in a strainer set over a bowl to catch the juices. Let drain for a few minutes, pressing on the pineapple gently, then return the juices to the pan, leaving the pineapple to drain further. Cook the juices over medium heat until reduced and thickened, about 5 minutes, adding any additionally drained juices to the pan halfway through. (The pineapple will continue to drain, but don't add any more of the juices to the thickened sauce. Rather, drink them, or save to drizzle over slices of the finished cake.) Remove the sauce from the heat, and add the butter, salt and vanilla, stirring to combine. (If baking the cake in the skillet, let the sauce cool. If baking the cake in a different pan, grease the pan, then pour the hot sauce into the pan to coat the bottom.)
Make the cake batter:
Position a rack in the lower-center of the oven and preheat to 350º.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl with a sturdy wooden spoon), cream the butter and sugars together on medium speed until lightened and fluffy, 5 minutes.
Meanwhile, sift or whisk together the flours, baking powder and salt into a smallish bowl. Stir the vanilla into the coconut milk.
Add the egg, then the egg white, to the creaming butter, beating after each addition until incorporated. Turn the mixer speed to low. Add 1/3 of the dry ingredients, beating until just incorporated, then half of the coconut milk. Repeat, then add the rest of the dries. Remove the bowl from the stand, and fold with a rubber spatula to make sure the batter is well-mixed.
Assemble and bake the cake:
Lay the cooked, drained pineapple chunks over the caramel. Spread the cake batter over the pineapple. Bake the cake until golden, beginning to pull away from the sides of the pan, and a tester comes out clean, 35 - 40 minutes. Let the cake cool for a few minutes, then run a thin knife around the sides of the cake to loosen it. Carefully invert the cake onto a large plate or serving platter. (To do this painlessly, invert a plate over the cake, then, wearing oven mitts, grasp the pan and plate together and flip them both over.) Remove the pan from the cake, replacing any pineapple chunks that may have become dislodged or stuck to the pan.
Let the cake cool to room temperature, 1 - 2 hours. Serve slices with scoops of coconut lemongrass ice cream and chopped, toasted macadamias. The cake is best served the day it is made, but will keep for a few days in the fridge; re-warm in an oven or toaster oven.
Make the Ice Cream:
In a small saucepan, heat the lemongrass, coconut milk and half the sugar until very warm, swirling occasionally. Turn off the heat, cover the pot and steep for 30 minutes. Strain through a fine mesh sieve, pressing on the lemongrass to extract all the coconut milk. Discard the lemongrass.
In a small bowl, whisk together the remaining sugar and the cornstarch. Whisk in the egg yolks. Dribble in the warm coconut milk, whisking constantly. Pour the mixture back into the saucepan, place over a medium-low flame, and cook, stirring and scraping the bottom of the pot constantly with a heat-proof spatula, until the mixture thickens and reaches 170ºRemove from the heat and immediately strain into a heatproof container. Stir in the heavy cream and the lime zest.
Chill the mixture until very cold, at least 4 hours and up to 2 days.
Spin in an ice cream maker, then chill in the freezer for 2 hours until firm.
Cake Adapted from Cook's Illustrated. Ice cream Adapted from Nordljus.Make the ice cream a day ahead (or early in the day), so that the base has time to chill, and the churned ice cream has time to 'cure' or firm up in the freezer.I cooked the pineapple and baked the cake in a cast iron skillet whose base measures 6 1/2" in diameter, and the top 8". I'd imagine you could bake the cake in a 6 or 7" cake pan with 2" high sides, or make individual cakes in well-greased muffin cups. You'll have just the right amount of coconut milk (and an egg yolk, too) leftover to make Coconut Lemongrass Ice Cream, below. Alternately, you can increase the cake recipe by 25% (1 cup of sugar in the topping, etc.), bake the cake in a 10" skillet or 9" cake pan, and serve the cake with store-bought (or home-made) vanilla ice cream. Complicated, I know. See? Pineapples are complicated.A ripe pineapple should smell tantalizing, and be mostly yellow in color. Adding half of a vanilla bean to the pineapple topping in place of the ginger could be a nice change.At work, we use pineapple peel to make Chicha Morada, an Andean fruit punch made with purple corn, citrus and spices: tasty.Nutritional values are based on one of six servings of cake.Timings are based on the cake only.