Preheat oven to 350°F. Line or prep a standard 12 count muffin tray. In a medium bowl, whisk together flour, cocoa, espresso powder, baking soda, and salt. Set aside.
With a stand mixer or electric beaters, cream together softened butter and brown sugar until fluffy (2 to 3 minutes).
Add in the egg and beat well. Add in the vanilla and sour cream, beat again, scraping sides if needed.
Beginning and ending with the dry mixture, add alternately with the cold brew coffee, stirring until just combined after each addition.
Scoop batter into prepared muffin liners and bake for 16 to 20 minutes or until tester comes out clean and cake tops spring back when touched.
Let cupcakes cool in pan for 5 minutes then remove to a cooling rack to cool completely before frosting. Follow with Coffee Buttercream (below).