Preheat oven to 175°C (350°F). Grease two 9-inch round cake pans, line the bottoms with parchment paper, and lightly dust the sides with flour.
Melt the unsweetened chocolate and set aside to cool for at least 10 minutes.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a large bowl, beat the softened butter and sugar together with an electric mixer until light and fluffy, about 3 minutes. Add the eggs one at a time, then mix in the vanilla extract.
Mix in the cooled melted chocolate until fully combined.
Add the dry ingredients and milk in alternating thirds, beginning and ending with the dry ingredients. Mix on low speed until just combined.
Divide the batter evenly between the two prepared pans. Bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean.
Cool in the pans for 10 minutes, then transfer to wire racks to cool completely.