Preheat oven to 350°F, line muffin tin with wrappers, set aside.
In a bowl, whisk together the 1 1/2 cup all-purpose flour, 3/4 cup white sugar, 1 teaspoon baking powder, 1 teaspoon baking soda and 1/4 teaspoon salt.
In another bowl, combine the 1 1/2 cup mashed bananas, 1/4 cup vegetable oil, 1 teaspoon vanilla extract and 1 egg.
Add the wet mixture to the dry and mix until JUST combined (don't overmix it)
Scoop batter into 12 cupcake liners. I used a large cookie scoop to do this, but aim for 2/3 full.
Bake for 18 minutes, or until they spring back when you touch them lightly.
Cool in the pan for a few minutes, then set on a wire cooling rack until completely cool. Frost.
Brown Sugar Buttercream Frosting
In a saucepan, whisk together the 1/3 cup all-purpose flour and 1 cup milk and cook over medium heat, whisking continuously until thickened to a smooth paste.
Remove from heat, stir in the 1 teaspoon vanilla extract and let cool to room temp.
Using a stand mixer fitted with the wire whisk, cream the 1 cup butter and 1 cup dark brown sugar until light and fluffy. There should be no grittiness.
Add in the milk mixture and beat it on high speed until you get a whipped cream consistency.
Frost cupcakes, and sprinkle with 1 tablespoon raw sugar.