Preheat the oven to 350°F. Line a 9 × 5-inch loaf pan with parchment paper and lightly grease the exposed sides.
Whisk together the flour, baking soda, baking powder, salt, and cinnamon in a medium bowl.
In a large bowl, whisk the melted, slightly cooled butter with the brown sugar. Add the eggs and vanilla and whisk until smooth.
Add the mashed bananas and Greek yogurt to the wet mixture and whisk until evenly combined.
Add the dry ingredients and gently fold with a rubber spatula until no obvious streaks of flour remain. Do not overmix.
Transfer the batter to the prepared loaf pan and smooth the top.
Bake at 350°F for 55–65 minutes.
The loaf is done when the top is golden brown, the center is set, and a toothpick inserted into the center comes out with a few moist crumbs but no wet batter. If the top browns too quickly, loosely tent it with foil.
Cool in the pan for 10–15 minutes.
Lift the loaf out using the parchment overhang and transfer it to a wire rack.
Let it cool before slicing.