Preheat the oven to 350°F (175°C). Line two 8x4-inch bread pans with parchment paper or grease them well with butter or baking spray. This ensures the banana bread will easily come out of the pans after baking.
In a medium bowl, use a fork to mash the bananas until smooth, leaving a few small bits for texture if desired. Set aside. I like them to be well mashed but still a little chunky for bursts of banana flavor in the bread.
In a separate large bowl, use a hand mixer or stand mixer to cream together the softened unsalted butter and white sugar until light and fluffy. Then, add the eggs one at a time and continue mixing until the mixture is well combined.
Add the all-purpose flour, baking soda, and table salt to the butter, sugar, and egg mixture. Mix on low speed until just combined, being careful not to overmix.
Gently fold the mashed bananas from Step 2 and the vanilla extract into the batter, mixing until just incorporated and the bananas are evenly distributed. I've found that folding by hand keeps the bread moist and tender.
Divide the batter evenly between the prepared bread pans from Step 1. Bake in the preheated oven for 35-40 minutes, or until the tops are puffed and browned and a cake tester inserted in the center comes out clean.
Let the breads cool in their pans for 5-10 minutes, then carefully remove them and transfer to a wire rack to finish cooling. Serve warm, at room temperature, or cold. I like to spread a little butter on a warm slice for extra indulgence!