Preheat the oven to 425 degrees F. Generously coat a mini doughnut pan with cooking spray. Whisk the flour, baking powder, baking soda, 1/2 teaspoon cinnamon and the salt in a large bowl. Whisk the mashed bananas, yogurt, vegetable oil, brown sugar, eggs and vanilla in a separate bowl. Stir the banana mixture into the flour mixture until just combined; stir in the chocolate chips.
Transfer the batter to a large resealable plastic bag and snip off one corner. Pipe the batter into the prepared pan, filling each cup almost all the way. Bake until golden brown, 7 to 8 minutes. Let cool in the pan2 to 3 minutes; remove the doughnuts to a rack to cool completely. Letthe pan cool, then coat with more cooking spray and repeat with the remaining batter.
Put the melted butter in a shallow bowl. Mix the granulated sugar and the remaining 1/4 teaspoon cinnamon in another shallow bowl. Dip one side of each doughnut in the butter, letting the excess drip off, then immediately dip it in the cinnamon sugar. Store in an airtight container up to 2 days.
You can also make these in a mini muffin pan; bake 8 to 10 minutes.