In a small pan, melt the butter over medium heat. As soon as the butter begins to bubble and foam, begin to swirl the pan. Swirl constantly until the foam subsides and the butter smells nutty and has turned a deep golden brown. Immediately pour butter into a heat-proof bowl and let it cool to room temperature.
Heat oven to 350° F. Line a standard muffin pan with liners. In a large bowl, whisk together eggs, sour cream, sweet potato, brown sugar, and vanilla. Whisk in cooled brown butter.
In a separate medium bowl, whisk together flour, baking powder, baking soda, cinnamon, salt, and nutmeg. Fold dry mixture into wet mixture until just combined. Fill muffin cups ¾ full and sprinkle each with 1 tablespoon chopped pecans. Bake until puffed and set, 20-22 minutes. Let cool slightly before serving.
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