Preheat oven to 350°F. Grease an 8x8x2-inch baking pan; set aside. In a medium bowl stir together the flour, granulated sugar, the 2 tablespoons cocoa powder, the baking powder, and salt. Stir in the milk, oil, and vanilla. Stir in the walnuts.
Pour batter into prepared baking pan. In a small bowl stir together the brown sugar and the 1/4 cup cocoa powder. Stir in the boiling water. Slowly pour brown sugar mixture over batter.
Bake in the preheated oven for 40 minutes. Transfer to a wire rack and cool for 45 to 60 minutes. Serve warm. Spoon cake into dessert bowls; spoon pudding from the bottom of the pan over cake. If desired, serve with vanilla ice cream.