Caramelize the Bananas: In a large skillet, melt ½ cup of unsalted butter over medium-high heat. Add the brown sugar, salt, ½ cup shredded coconut, and lemon juice; swirl the pan until melted and combined. Add the thick banana slices and swirl to coat (do not stir, or they will turn to mush!). If using rum, pour it in now and cook just until bubbling. Remove from heat and set aside to cool.
Whip the Eggs: Preheat the oven to 350°F (177°C). In the bowl of a stand mixer fitted with the whisk attachment, beat the room-temperature eggs, vanilla sugar, white sugar, and salt on high speed until the mixture has doubled in volume and turns a pale, fluffy yellow.
Mix the Batter: In a separate bowl, whisk together the flour, cornstarch, and baking powder. With the mixer on low speed, gradually add the dry ingredients to the eggs and mix until just combined. Pour in the melted butter and mix or fold with a spatula only until incorporated.
Layer the Pan: Do not grease your pan! Pour the cooled, caramelized banana mixture directly into the bottom of a clean, ungreased 9-inch non-stick bundt pan, swirling slightly to create an even layer. Carefully scoop and gently spread the cake batter over the bananas without disturbing the fruit layer below.
Bake and Invert: Bake on the middle rack for 25 minutes, or until a wooden skewer inserted into the cake comes out clean. Let the cake cool in the pan for 10 minutes. Run a wooden skewer around the outer edges to loosen it the cake layer. Place a serving platter upside down over the pan, grip them tightly together, and quickly flip!
Garnish and Serve: Let the inverted cake cool for another 5 to 10 minutes. Dust evenly with powdered icing sugar and serve warm!