Pour rum over ground walnuts in a small bowl; soak for 1 hour.
Grease and flour a 9-inch log mold.
Mix 1/3 cup butter and 1/2 cup plus 2 1/2 tablespoons sugar together with an electric mixer in a large bowl; beat in egg yolks until light and fluffy. Add milk, flour, cocoa powder, and baking powder; mix until smooth a batter forms.
Whisk egg whites in a separate bowl until soft peaks form. Lift your beater or whisk straight up: the egg whites will form soft mounds.
Spoon 1/3 of the egg whites into batter using a rubber spatula. Gently run spatula around the edges of the mixture lifting the bottom up and over the center, repeating until well incorporated. Add remaining egg whites, folding just until blended.
Pour batter into prepared log mold and place in cold oven (not preheated).
Set oven to 300 degrees F (150 degrees C); bake for 15 minutes. Increase oven to 375 degrees F (190 degrees C); bake until top springs back when lightly pressed, 25 to 30 minutes more.
Cool cake in the pan for 15 minutes before transferring to a wire rack to cool completely, about 30 minutes more.
Blend 3/4 cup plus 2 tablespoons butter and 1/2 cup sugar together with walnut and rum mixture. Transfer cooled cake to serving platter; cut in half lengthwise. Spread walnut mixture evenly over bottom half of cake; replace cake top; slice vertically to serve.