Heat the oven to 360 °F. Give the oven time to fully preheat so the batter begins rising as soon as it goes into the pan.
Butter three 7x3x2 mini loaf pans or one 9×5-inch loaf pan. Coat the corners and sides well so the baked loaf releases without tearing.
Beat the bananas, brown sugar, and vanilla in a mixing bowl with a hand held mixer or a stand mixer fitted with the paddle attachment for 30 seconds to 1 minute. The mixture should look glossy and aerated, with a few small banana lumps still visible.
Mix in the eggs one at a time, allowing each egg to disappear before adding the next, then blend in the sourdough starter. The batter should look cohesive and thick, without separate streaks of egg or starter.
Sift the flour, baking soda, and salt into a separate bowl, then fold the dry mixture into the banana batter in batches. Stop each fold when the flour begins to disappear, leaving the batter soft with gentle resistance.
Add the milk and neutral flavored oil, then mix only until the batter comes together. Avoid continued mixing after the last dry pockets vanish, because extra mixing can make the crumb tough.
Scrape the batter into the prepared pan or pans, setting mini loaf pans on a baking sheet before placing them in the oven. The batter should sit evenly in the pan, ready to rise into a rounded top.
Bake mini loaves for 45 minutes or the 9×5-inch loaf for 60 to 65 minutes. The crust should be rich golden brown and a tester should come out without wet batter; loosely cover with foil if the top darkens too quickly.
Leave the bread in its pan for 20 minutes before moving it to a wire rack. This pause lets the fragile interior set so the loaf is less likely to crack or collapse during transfer.
Cool the loaf completely, then finish with powdered sugar if desired. Wait until the crumb is no longer steaming before slicing for the cleanest, least gummy pieces.