Preheat the oven to 350 degrees F. Line a 12-cup muffin pan with cupcake liners.
Melt the butter in a small skillet set over medium heat, and cook until it starts to bubble and foam, about 5 minutes. Continue cooking, stirring constantly, until the foam subsides and brown specks start to appear, 1 to 2 minutes. Transfer the butter to a medium mixing bowl to cool slightly.
In a large bowl, combine the flour, 1/2 cup of the oats, the baking powder, salt and baking soda.
Add the sour cream, sugar and vanilla to the browned butter and whisk. Add the eggs and whisk until combined. Fold the wet ingredients into the dry ingredients and mix until just combined. Then fold in the bananas (don't worry if there are a few lumps).
Divide the batter evenly among the prepared pan. Sprinkle with the remaining 2 tablespoons oats. Bake until the muffins are light golden and a toothpick inserted in the center comes out clean, 20 to 24 minutes. Cool in the pan for a few minutes, and then transfer to a wire rack to cool completely.
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