To make the chocolate donuts: Place a rack in the center of the oven and preheat to 375 degrees.
In a medium mixing bowl, sift together the flour, cocoa powder, and baking soda. Add the brown sugar and espresso powder, whisk and set aside. In a small bowl, beat together the vanilla, the egg, sour cream, milk, and brown butter. Stir the wet ingredients into the dry until just combined, do not overmix. Spoon the mixture into a greased donut pan. Bake donuts for 8 minutes or until the tops spring back at the touch. Let donuts cool in pan before dipping in the glaze.