Melt the butter in a small, heavy saucepan over medium-low. Increase heat to medium-high, and cook, stirring occasionally, just until the butter is browned and smells nutty, 4 to 5 minutes. Pour into a metal bowl, and let cool to room temperature, about 20 minutes.
Whisk together flour, sugar, baking powder, baking soda, and salt in a large bowl. Combine sour cream, eggs, cinnamon, vanilla, and brown butter in a separate bowl. Stir butter mixture into flour mixture until combined. Fold in rhubarb.
Place baking-cup liners in a 12-cup muffin pan. Spoon a heaping ⅓ cupful into each muffin cup. Bake in oven until a wooden toothpick inserted in center comes out clean, 18 to 22 minutes. Cool in pan 5 minutes; remove muffins to a wire rack and cool completely, about 20 minutes.
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