In a bowl, whisk together egg, vanilla, milk and yogurt. Set aside.
In a mixing bowl, mix together flour, sugar, cocoa powder, baking powder, baking soda and salt.
Then, add in butter.
Gradually, add in egg mixture and beat until combined.
Let dough rest for 15 minutes.
On a well floured surface, turn dough out. Roll dough out to approx 1/2 inch thick. Cut out dough with doughnut cutter or glass/lid.
In a large, heat proof pot, pour in oil. You will need to have approx 2-3 inches of oil in the pot. Heat oil until it reaches 360-375 degrees.
Drop doughnuts and doughnut holes into hot oil. Work in batches of about 3-4 doughnuts. Do not overcrowd your pot.
Cook each for only 1-2 minutes on each side.
Remove from hot oil and let doughnuts drain on a plate lined well with paper towels.
Let doughnuts cool before adding glaze.