In a bowl, combine flour, sugar, baking powder, baking soda, cinnamon and salt. In another bowl, combine mashed bananas, eggs, oil, milk, and vanilla.
Pour the wet ingredients into the dry and whisk until well combined. Set aside.
In a skillet, melt butter over medium heat. Stir in brown sugar and rum and allow to simmer. Remove from heat.
Preheat oven to 350F. Grease a donut baking pan with cooking spray. Drizzle a small amount of the sugar mixture into each well.
Arrange the banana slices on top of the sugar mixture, in a single circular pattern.
Spoon the batter on top of the bananas, filling the donut wells 3/4 of the way full. Bake for about 15-20 minutes, or until an inserted toothpick comes out clean.
Cool for about 8 minutes, and then flip over the pan to release the donuts. Optional (but totally awesome): pour a little more rum on the donuts and carefully flambé!