Preheat the oven to 325 degrees F and arrange the rack in the center. Lightly grease and flour a 9x5-inch loaf pan. Remove the peels and place the bananas in a small bowl. Thoroughly mash the bananas with a fork. Melt the unsalted butter and allow it to return back to room temperature.
Use a stand mixer fitted with the whisk attachment to beat together the brown sugar, white sugar, and eggs on medium speed until pale and fluffy, about 3-5 minutes.
Switch the mixer speed to low and slowly drizzle in the oil, then drizzle in the butter (making sure it's cooled down from being melted). Add the mashed bananas, yogurt or sour cream, and vanilla extract. Continue to mix at low speed until just combined.
In another large bowl, mix together the flour, baking soda, cinnamon, and salt.
Add all of the dry ingredients to the wet ingredients and using a spatula (don't use the mixer), gently fold in the dry ingredients until just combined. Don't overmix this batter; just fold until you don't see any streaks of the dry ingredients.
Pour the batter into the prepared loaf pan. If desired, sprinkle the 2 tablespoons of brown sugar and 1 tablespoon white sugar evenly over the top of the bread.
Place the bread on a baking sheet. Bake for 50-70 minutes (wide variance, depending on the actual pan you used, actual heat of your oven, humidity, etc.), until lightly golden brown on top, and bread springs back when you press it, and a toothpick, when inserted in the center, comes out clean or with moist crumbs.
Remove the bread and allow it to cool in the pan for 8-10 minutes. Invert onto a cooling rack to finish cooling and then slice with a sharp serrated knife.