For the ice cream: Prepare an ice water bath. Bring the cream and coconut milk to a boil in a medium saucepan. Whisk together the egg yolks and sugar in a bowl; slowly pour in some of the hot cream to temper the yolks. Pour the eggs into the cream mixture and cook over medium-low heat, stirring constantly, for 2 to 3 minutes, or until the custard coats the back of a spoon and steam rises from the top. Strain the custard through a fine-mesh sieve into a clean bowl and cool over the ice water bath, stirring occasionally, until the custard is chilled. Process the custard in an ice cream machine. Fold in the toasted coconut and keep the ice cream frozen until ready to use.

For the pudding cake: Preheat the oven to 350 degrees F. Line an 8-inch square pan with plastic wrap. Whisk together the flour, 3/4 cup of the brown sugar, 1/4 cup of the cocoa powder, and the baking powder. Fold in the cream and pour the batter into the plastic-lined pan. Mix the remaining 3/4 cup brown sugar and 1/4 cup cocoa powder together and sprinkle over the top of the batter. Bring the coconut water to a boil and pour over the cake batter. Place the cake pan in a large roasting pan; pour enough water into the roasting pan to come 1-inch up the side of the cake pan. Bake the cake in the water bath for 50 minutes. Cool the cake to room temperature and invert it onto a sheet pan. Carefully pull away the plastic wrap. There should be a gooey pudding-like substance on the bottom of the cake. Using a 4-inch diameter ring cutter, cut 4 portions from the cake. Clean the ring cutter before each cut.

For the emulsion: Warm the coconut milk and sugar in a small saucepan. Just prior to use, froth with a handheld blender.

Assembly: Place a piece of the pudding cake in the center of each plate. Spoon the frothy portion of the emulsion around the pudding cakes. Place a quenelle-shaped scoop of the ice cream next to the cake and sprinkle the toasted coconut, coquito nuts, and crystallized ginger over the ice cream and around the cake.