1. To make the topping, place the brown sugar and water or butter in an 8-inch (20 cm) square cake pan. Warm the pan directly on the stovetop over low heat, stirring until the sugar is thoroughly moistened.
If using water, simmer the mixture for about 45 seconds. If using butter, stir just until the sugar is moist and bubbling, then remove from heat. (It won’t melt completely smooth, and there may be a few bare spots, which is normal.) Let cool to room temperature.
2. Peel and slice the bananas in 1/4-inch (1 cm) slices. Arrange them in slightly overlapping rows over the melted brown sugar. Sprinkle with a few drops of lemon juice.
3. Preheat the oven to 350ºF (180ºC).
4. Whisk together the flour, baking powder, baking soda, salt, and cinnamon in a large bowl, making sure there are no lumps. Mix in the granulated sugar.
5. In a small bowl, mix together the butter, egg, egg white, banana puree, sour cream, and vanilla.
6. Make a well in the center of the dry ingredients and stir in the wet ingredients until almost combined. Do not overmix. Gently fold in the chocolate pieces.
7. Scrape the batter into the pan over the bananas, then use a spatula to carefully spread the batter over the sliced fruit.
8. Bake for 40 minutes, or until the cake feels just set in the center when you touch it.
9. Cool the cake for about 20 minutes, then run a knife along the edges of the cake to help it release from the pan. Invert the cake onto a serving platter.