Begin by preheating your oven to 350°F. Prepare a 12-cup muffin tin by either lining it with paper liners or generously greasing each cup with butter or cooking spray.

In a large mixing bowl, mash the bananas until smooth. Stir in the sugar and egg, mixing until everything is well incorporated and smooth. Next, blend in the vegetable oil until fully combined.

Incorporate the flour, baking soda, baking powder, cinnamon, salt, and nutmeg (if using). Fold in 3/4 cup of the chopped walnuts as you add the dry ingredients. This technique ensures the walnuts are evenly dispersed and helps avoid over-mixing the batter.

In a separate, smaller bowl, mix together the remaining 3/4 cup walnuts, brown sugar, and melted butter, making sure the walnuts are evenly coated.

Evenly divide the muffin batter among the 12 prepared muffin cups. Top each muffin with a portion of the walnut topping, spreading it across the surface of each muffin so that it doesn't clump together, which might cause it to sink during baking.

Bake the muffins for 20 to 23 minutes, or until they are golden brown and a toothpick inserted into the center of a muffin comes out clean. Allow the muffins to cool in the tin for a few minutes before transferring them to a wire rack to cool completely.