Heat oven to 350°F and line a 12-cup muffin tin with paper liners. Line a second muffin tin with 3 more liners. This recipe makes about 15 cupcakes.
Cut the stick of butter into 5-6 pieces. Melt the butter pieces in a light-colored skillet (if you have one) over medium heat. Keep cooking, stirring constantly, until brown bits start forming on the bottom of the skillet and the butter smells toasty. This can take anywhere from 3-8 minutes. Scrape into a heatproof bowl right away and set aside.
Combine 1 3/4 cups flour, 1 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/2 teaspoon salt in a large bowl with a whisk.
In a separate large bowl, mash the 4 bananas with a potato masher or fork. If the bananas have been frozen and are still cold, let them come to room temperature before adding the brown butter to prevent the butter from seizing up and hardening. Then whisk in brown butter, 3/4 cup sugar, 2 eggs, 1/4 cup sour cream, and 1 tablespoon vanilla. Whisk together wet ingredients until smooth with a few banana pieces here and there.
Add the flour mixture to the wet ingredients and gently stir with a rubber spatula just until evenly combined, do not over mix. Spoon batter into muffin cups with a spoon or a cookie scoop. Fill about 2/3 full.
Bake until a toothpick inserted in the center of a cupcake comes out clean, about 25 to 30 minutes. Let cool on a wire rack for 5 minutes then remove from pan and let cool completely before frosting.