Preheat oven to 350 degrees. Butter a 9-inch (5-cup) trois freres mold or Bundt pan. Dust with flour, and tap out excess. Put butter and brown sugar into the bowl of an electric mixer fitted with the paddle attachment. Mix on medium-high speed until pale and fluffy, about 3 minutes.
Add eggs, one at a time, mixing well after each addition. Add vanilla and carrots, and mix until just combined. Reduce speed to low. Add flour, baking powder, salt, and baking soda. Mix until just combined. Stir in walnuts.
Pour batter into the prepared mold. Bake until a cake tester inserted into center comes out clean, about 30 minutes. Let cool in pan on a wire rack 15 minutes. Turn out cake onto rack, and let cool completely. Before serving, dust with confectioners' sugar. Cake can be stored in an airtight container at room temperature up to 3 days.