Into a small mixing bowl, add Active Dry Yeast (1 3/4 teaspoon), Granulated Sugar (1 teaspoon), and Water (6 tablespoon). Don't mix. Let sit for 5 minutes.
Into the bowl of a stand mixer fitted with a paddle attachment, add All-Purpose Flour (1 1/2 cup), Granulated Sugar (3 tablespoon), Salt (1 teaspoon), yeast mixture, Buttermilk (2/3 cup), and Egg (1). Beat on medium speed for 1 minute.
Add Unsalted Butter (2 tablespoon). Mix to combine.
Add remaining All-Purpose Flour (1 1/2 cup) a bit at a time until mixture comes together into a dough, switching paddle attachment to a dough hook when needed.
When dough pulls away from the sides of the bowl but is still sticky, remove it to a greased bowl. Place the bowl in a pan of boiling water and keep it in the oven to rise for 1 hour.
Turn dough out onto a floured surface. Knead a few times to press all the air out. Roll out into a large rectangle.
Cut out your doughnuts. Place them on a baking tray lined with parchment paper. Place in the oven to rise for another 30 minutes.
In a pot, heat Cooking Oil (as needed) to 345 degrees F (174 degrees C). Add doughnuts to the oil and cook for about 1 minute on each side, until light golden brown. Remove to a tray lined with paper towels to drain excess oil.
Into a mixing bowl, add Powdered Confectioners Sugar (2 1/2 cup), Butter (4 tablespoon), Unsweetened Cocoa Powder (3 tablespoon), Milk (3 tablespoon), and Vanilla Extract (1/4 teaspoon). Whisk until smooth.
Coat doughnuts in the chocolate glaze and serve!