Preheat the oven to 350F with rack in the center of the oven.
Combine the light brown sugar and white sugar in a large mixing bowl.
Remove ½ cup of that sugar mixture to a small bowl and stir in the 2 teaspoons cinnamon.
Butter a loaf pan. Sprinkle a few tablespoons of the cinnamon-sugar mixture inside the buttered loaf pan. Set aside the loaf pan and the rest of the cinnamon sugar.
In the large bowl with the ¾ cup of sugar mixture, mash in the bananas until mostly smooth (small chunks okay). Whisk in the melted butter, eggs, yogurt, and vanilla.
In a small bowl, combine the flour, baking soda, and salt.
Stir the dry ingredients into the wet.
Spoon half of the batter into the sugar-coated loaf pan. Sprinkle half of the remaining cinnamon sugar mixture over the batter. Spoon the rest of the batter on top and cover with the remaining cinnamon sugar.
Bake for 60 minutes, or until a skewer/toothpick poked in the center comes out clean. Allow to cool on a cooling rack for 15 minutes before removing from loaf pan to cool completely on the rack (or as long as you can wait until slicing).