Preheat oven to 350°F (175°C). Spray an 8×11-inch baking dish (or a similar-sized dish) with nonstick spray and set aside.
In a medium bowl, whisk together 2 1/2 cups all-purpose flour, 2 teaspoons baking powder, 1/4 teaspoon cinnamon, and 1/4 teaspoon salt. Set the dry mixture aside.
In a clean bowl, use an electric mixer or hand mixer to beat the 2 egg whites until stiff peaks form. Transfer the beaten egg whites to a separate bowl and set aside.
In the mixer bowl (or another large bowl), combine the 2 egg yolks, 1/2 cup brown sugar, and 1/4 cup granulated sugar. Beat on medium speed until the mixture is creamy and well combined.
Add 2 teaspoons vanilla extract and the 4 bananas, mashed (about 1 1/4 cups) to the yolk-sugar mixture and mix on low until incorporated.
With the mixer on low, add the cooled 1/2 cup (1 stick) brown butter and mix just until combined.
Add half of the dry flour mixture to the batter and mix briefly on low. Pour in the 2/3 cup milk (whole, 2%, soy, or almond) and mix on low. Add the remaining dry ingredients and mix until just combined. Do not overmix.
Remove the bowl from the mixer and gently fold the beaten egg whites into the batter with a large spatula until no large streaks of egg white remain.
Pour the batter into the prepared baking dish, smoothing the top. Bake for 25–30 minutes, or until the cake is set in the center and slightly golden. A tester inserted into the center should come out with a few moist crumbs but not raw batter.
Remove the cake from the oven and let it cool slightly in the pan for 10–15 minutes. You may leave the cake in the pan or carefully invert it onto a rimmed platter—either is fine. Let the cake cool completely.
Once the cake is completely cool, use a fork to poke many holes all over the top of the cake.
In a large measuring cup or bowl, whisk together 1/2 cup light coconut milk, 1/2 cup sweetened condensed milk, and 1/2 cup evaporated milk. Pour the milk mixture evenly over the cake, allowing it to seep into the holes. Refrigerate the cake and let it set for about 45 minutes.
To make the whipped topping, add 1 1/2 cups cold heavy whipping cream to a chilled mixing bowl and beat until stiff peaks form (about 3–4 minutes). Add 2 tablespoons powdered sugar and beat briefly on medium just to combine. Spread the whipped cream evenly over the soaked cake.
To make the caramelized bananas, heat a saucepan over medium heat. Add 3 tablespoons unsalted butter and let it melt and begin to sizzle. Add the 3 firm, sliced bananas, 3 tablespoons brown sugar, 1/4 teaspoon cinnamon, and a pinch of salt. Cook, gently tossing or stirring occasionally, for about 4–5 minutes until the bananas are golden and coated in caramel. Remove from heat and let cool slightly.
Spoon the caramelized bananas over the top of the cake or serve them alongside as a garnish. Store the cake in the refrigerator. If desired, let refrigerated slices sit at room temperature for 30–60 minutes before serving for a softer texture.