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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.
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Add the sugars and continue to beat for another 3 minutes.
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Add the eggs one by one, beating for 1 minute after each addition.
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Beat in the vanilla; don't be concerned if the mixture looks curdled.
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Reduce the mixer speed to low and mix in the melted chocolate.
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When chocolate 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.
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Still working on low speed, mix in the boiling water, which will thin the batter considerably.
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Switch to a rubber spatula, scrape down the bowl and stir in the chopped chocolate.
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Divide the batter evenly between the prepared liners, filling each about 3/4 full. Bake until a toothpick inserted in the center comes out clean, about 14-16 minutes, rotating the pans halfway through baking.
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Let cool in the pan about 2-3 minutes, then transfer to a wire rack to cool completely.