Assemble the cake: Trim 1/2 inch off the short ends of the pound cake, then trim the domed top to make it flat. Stand the cake on its side and slice in half horizontally to make two equal-size rectangles.
Microwave the chocolate in 30-second intervals, stirring until melted. Beat the butter, 3 tablespoons milk and the confectioners' sugar with a mixer. Beat in the melted chocolate. Add more milk, if needed, to make the frosting spreadable.
Lay the cake rectangles on a rack set on a rimmed baking sheet. Spread some of the frosting on one short end of each cake and press together to form a long rectangle. (You can slide a thin piece of cardboard under the cake for moving to a platter later.)
Spread the remaining frosting on the cake in a 1-inch-thick layer, making the edges slightly higher than the center. Smooth the top and sides with an offset spatula. Freeze until firm, about 30 minutes.
Mix the peanuts and dulce de leche in a bowl.
Remove the cake from the freezer. Spread the peanut mixture on the frosting in a flat, even layer. Freeze until the dulce de leche is firm, about 30 minutes.
Meanwhile, make the chocolate coating: Microwave the chocolate and butter in 30-second intervals, stirring, until melted and smooth.
Pour the chocolate on the cake and spread it evenly over the top and sides with an offset spatula. Freeze until the chocolate cools slightly, 6 to 8 minutes.
Starting at a short end of the cake, dip the edge of an offset spatula into the chocolate at an angle and gently pull up, repeating along the top of the cake to create a wave pattern. Chill 10 minutes before serving.