In the bowl of a stand mixer, combine the sourdough starter, warm water, brown sugar, bread flour, Dutch-process cocoa powder, espresso powder (if using), and salt.
Using the dough hook attachment, mix on low speed until the dough comes together. As the mixer continues running, add the softened butter one tablespoon at a time, allowing each addition to fully incorporate before adding the next.
Continue kneading on medium speed for several minutes, then increase the mixer to high speed and continue kneading for a total of 7–10 minutes, or until the dough is smooth, elastic, and no longer sticking to the sides of the bowl.
No stand mixer? Mix everything together in a large bowl, then knead by hand for 10–12 minutes, gradually working the softened butter into the dough until fully incorporated.