Preheat the oven to 350°F. Butter and flour two 8-inch round cake pans, then line the bottoms with parchment paper.

In a medium bowl, whisk the all-purpose flour, baking powder, and salt together until evenly distributed; set aside.

Cream the softened butter and granulated sugar in a stand mixer fitted with the paddle attachment on medium-high speed until pale and fluffy, about 3 minutes. Add eggs one at a time, mixing thoroughly after each, then beat in the vanilla.

With the mixer on low, add the flour mixture and milk in alternating batches, starting and ending with the flour, using three additions of flour and two of milk. Mix just until each addition is incorporated.

Divide the batter evenly between the prepared pans. Bake for 35 to 40 minutes, until a toothpick inserted in the center comes out mostly clean.

Let the layers cool in the pans on a wire rack for 10 minutes, then turn them out onto the rack to cool completely.

In a large bowl, whisk together the sifted powdered sugar and Dutch-processed cocoa powder.

Beat the butter until smooth and creamy, then gradually beat in the sugar-cocoa mixture about 1/2 cup at a time, beating well after each addition. Continue beating until light and smooth, approximately 5 minutes.

Pour in the cooled melted chocolate and beat until fully combined. Add milk a tablespoon at a time until the frosting reaches your desired spreading consistency.

Place one cake layer on a serving plate or cake stand and spread about 1 cup of frosting over the top. Set the second layer on top, then apply a thin crumb coat over the entire cake. Refrigerate for 20 minutes.

Frost the top and sides with the remaining buttercream using an offset spatula, leaving a rustic finish.

Finish the top with rainbow sprinkles and decorative birthday flags if you like.