In a bowl of a stand mixer, mix together the flour, cocoa powder, espresso powder, and yeast, until well combined. Mix in the sugar and salt. Add the milk, eggs, and vanilla paste. Mix until the ingredients incorporate and continue mixing on low speed for 30 minutes.
Add the butter a bit at a time. Scrape the sides of the bowl to make sure everything is well incorporated. Turn the dough over onto a lightly floured surface and pat it into a rectangle shape.
Stretch and gently pull the left side of the dough and fold it over two-thirds of the dough. Do the same for the right side, as if you're folding a letter. Repeat this again for the bottom and top portion of the rectangle. Turn the dough over so that the seam side is down. Place it into a lightly greased bowl. Cover and let it sit at room temperature for one hour.
Turn the dough back over onto a lightly floured surface. Gently press out any bubbles. Repeat the stretch and folding process. Turn it over so the seam side is down. Cover and refrigerate overnight.
Roll the dough out into a 11" circle on a lightly floured surface. Transfer to a parchment lined sheet pan and refrigerate for 30 minutes or until the dough is firm enough to cut.
Cut 8 rounds from the dough with a 3 1/2" cutter. Cut 1 1/2" circles in the center. Place all cut dough onto a parchment lined sheet pan. Cover and let it proof for 1 - 1 1/2 hours until doubled in size.
Heat 3" of oil in a cast iron pot, or any heavy pot, to 350F. Set a cooling rack over a sheet pan. In a large bowl, combine the sugar, vanilla, and cocoa powder. Mix to combine and set aside.
Once the oil is hot, gently place the 3-4 doughnuts in the oil. Fry on one side for 30 seconds and flip it to cook for another 45 seconds. Flip it again and fry for another 45 seconds. Transfer the dough to the cooling rack. Toss the donuts in the sugar mixture and continue frying the rest.