Arrange a rack in the bottom third of the oven and heat to 350°F
Melt the butter in the microwave or over low heat on the stovetop.
Place the melted butter and sugar in a large bowl and whisk until combined.
Crack the eggs into the bowl. Whisk until completely combined and the mixture is smooth.
Whisk the milk and vanilla into the batter.
Peel the bananas and add them to the bowl. Using the end of the whisk or a dinner fork, mash them into the batter. Leave the bananas as chunky or as smooth as you prefer.
Measure the flour, baking soda, and salt into the bowl. Switch to using a spatula and gently stir until the ingredients are just barely combined and no more dry flour is visible.
Pour the batter into the prepared loaf pan, using the spatula to scrape all the batter from the bowl. Smooth the top of the batter.
Bake until the top of the cake is caramelized dark brown with some yellow interior peeking through and a toothpick or cake tester inserted into the middle comes out clean, 50 to 65 minutes. Baking time will vary slightly depending on the moisture and sugar content of your bananas — start checking around 50 minutes and then every 5 minutes after.
Set the loaf, still in the pan, on a wire cooling rack. Let it cool for 10 minutes — this helps the loaf solidify and makes it easier to remove from the pan.
lift the loaf out of the pan and place it on the cooling rack. Cool for another 10 minutes before slicing.