In a medium glass bowl, melt the butter and the chocolate together over a pot of barely simmering water. Remove from the heat and let cool slightly.

In another large bowl, whisk together the flour, sugar, cocoa, baking powder, baking soda, and salt. Add the buttermilk and vanilla to the melted butter mixture and stir to combine. Stir in the egg. Add the wet ingredients to the dry ingredients and stir with a wooden spoon until the mixture is combined. Transfer the dough to a piece of plastic wrap, wrap it well, and let it rest for 30 minutes in the refrigerator.

In a dutch oven with a candy thermometer attached, heat 3 inches of oil to 360° F. Scoop the dough into 1-tablespoon balls. Roll each ball into a circle.

Use a slotted spoon to gently lower the dough balls into the oil, 6 at a time. Adjust the heat as needed to ensure the temperature stays around 350° F and 360° F. Cook the donut holes until puffed and cooked through, 2 to 3 minutes. Use the slotted spoon to transfer the donuts to a paper towel-lined baking sheet. Repeat with the remaining dough.

In a small bowl, whisk together the confectioners’ sugar, milk, and vanilla until smooth. Add a little bit of water if necessary. Dip each donut into the glaze and transfer to a drying rack over a baking sheet so any excess glaze can drip off. Let the donuts stand until the glaze is no longer wet.