Working with a stand mixer, preferably fitted with a paddle attachment, or with a hand mixer in a large bowl, beat the butter on medium speed until soft and creamy. Add the sugars and continue to beat for another 3 minutes. Add the eggs one by one, beating for 1 minute after each addition. Beat in the vanilla; don’t be concerned if the mixture looks curdled. Reduce the mixer speed to low and mix in the melted chocolate. When it is fully incorporated, add the dry ingredients alternately with the buttermilk, adding the dry ingredients in 3 additions and the milk in 2 (begin and end with the dry ingredients); scrape down the sides of the bowl as needed and mix only until the ingredients disappear into the batter. At this point, the batter will be thick, like frosting. Still working on low speed, mix in the boiling water, which will thin the batter considerably. Switch to a rubber spatula, scrape down the bowl and stir in the chopped chocolate. Divide the batter evenly between the two pans and smooth the tops with the rubber spatula.
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