Cook butter in a saucepan over medium heat, stirring constantly until it foams, turns clear, and then turns a deep brown, about 6 minutes. Pour browned butter into a large mixing bowl and cool to room temperature.
Whisk together brown butter, buttermilk, yeast, salt, eggs, and brown sugar until combined. Add flour and mix in with a wooden spoon or dough whisk until incorporated. Cover bowl with plastic and let rest at room temperature for two hours.
When ready to bake, line two rimmed baking sheets with parchment paper and flour board or counter well. Scrape dough onto board and dust dough with flour and use a lightly floured rolling pin to roll dough into a ½-inch thick rectangle.
Use a floured 3-inch cutter to form 14 rounds; cut out the center of each round with a floured 1½-inch cutter. Place doughnuts on prepared baking sheets and let rest at room temperature for 45 minutes.
Preheat oven to 350°F
Bake doughnuts for 18 minutes, until golden. Place on a wire rack to cool for 5 minutes.
For the glaze, mix powdered sugar, salt, espresso powder and milk until smooth. Set rack with doughnuts over a piece of wax paper. Dip doughnut tops into glaze.
For the chocolate drizzle, put chocolate (chopped or chips) into a microwave safe bowl. Microwave in 30 second increments, stirring to be sure not to overheat chocolate. Microwave until melted, about 1 minute 30 seconds, depending on microwave. Stir in vegetable oil until chocolate mixture is a thin glaze.
Drizzle chocolate over glazed donuts and sprinkle chopped nuts over chocolate, before it sets.