Position a rack in the center of the oven and preheat to 325°. 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 tablespoons 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.