Preheat the oven to 425°F. Line a standard 12-cup muffin pan with paper liners.
Spread the walnuts on a small baking sheet and toast for 5 to 7 minutes until fragrant. Allow them to cool, then roughly chop.
In a large mixing bowl, mash the bananas until mostly smooth with a few small lumps remaining.
Whisk in the melted butter, brown sugar, granulated sugar, eggs, vanilla extract, and sour cream until fully combined.
In a separate bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, and nutmeg.
Add the dry ingredients to the wet ingredients and gently fold together until almost combined.
Fold in the toasted walnuts just until evenly distributed. Be careful not to overmix.
Divide the batter evenly among the muffin cups, filling each about ¾ to ⅞ full. Sprinkle the tops with coarse sugar if desired.
Bake for 5 minutes at 425°F. Without opening the oven door, reduce the temperature to 350°F and continue baking for 13 to 17 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the muffins cool in the pan for 5 minutes before transferring them to a wire rack to cool completely.