Position a rack in the center of the oven and preheat to 325°F Grease a 5-by-9-inch loaf pan. Trim both ends off each of 3 bananas to fit crosswise in the pan; reserve the ends. Halve the long banana pieces lengthwise; peel and transfer to a plate.
In a large, heavy skillet, bring 1/4 cup sugar and 2 tablespoon water to a boil, without stirring. Cook until the water has evaporated and the mixture starts turning amber. Remove from the heat and swirl in the butter with a fork. Add the long banana pieces cut side down and cook over low heat for 1 minute; transfer to the prepared loaf pan, arranging them crosswise, cut side down. Drizzle the remaining caramel on top.
In a medium bowl, mash the banana end pieces to equal 1 cup (add more banana if necessary). Mash in the remaining 1/2 cup sugar. Beat in the whole egg, egg white and vanilla.
In a large bowl, whisk the flour with the baking powder, cinnamon and salt. Add the banana mixture and oil, whisk until blended. Scrape the batter over the bananas in the pan and bake until a toothpick inserted in the center comes out dry, 50 to 55 minutes. Let cool on a rack for 10 minutes; invert the cake onto a platter and let cool completely.