Arrange a rack in the middle of the oven and heat the oven to 350°F. Coat the wells of a standard 12-well muffin pan with cooking spray or line with paper liners.
Cut 1 stick unsalted butter into 8 pieces and place in a large microwave-safe bowl. Microwave in 20-second bursts until mostly melted, 40 to 60 seconds total. Set aside to let the butter finish melting and cool slightly. (Alternatively, melt the butter on the stovetop, then pour into a large bowl.)
Peel and place 4 large bananas in the bowl of butter and use a fork or potato masher to mash into a purée. A few small lumps of banana are OK. Add 2 large eggs, ¾ cup packed light or dark brown sugar, 1 teaspoon vanilla extract, and ½ teaspoon kosher salt. Whisk until combined.
Sprinkle 1 teaspoon baking soda evenly over the banana mixture. Add 1 ½ cups all-purpose flour and stir with a flexible spatula until just incorporated and no dry spots remain.
Divide the batter among the prepared muffin wells, filling each about three-quarters full (scant ⅓ cup per well).
Bake, rotating the pan halfway through baking, until the muffins are golden, the tops bounce back when pressed, and a toothpick inserted into the center of a muffin comes out clean, 18 to 20 minutes total. Let the muffins cool in the pan for 5 minutes, then transfer to a wire rack and let cool completely.
