Preheat oil in a pan over low-medium heat (350°F - 1” oil deep)
In a large bowl, combine: 3/4 cup mashed potatoes (w/milk, butter & salt in it), 1/4 cup sugar and 1/2 tsp vanilla extract. Then add 1 egg and 3/4 cup sour cream. Stir wet ingredients until combined.
In a separate bowl, combine: 1 3/4 cups all-purpose flour, 1/2 tsp baking soda and 1/2 tsp baking powder. Sift the dry ingredients into the batter; stir until incorporated. Batter will be sticky.
Lightly oil large cutting board. Dipping hands into flour, make 1” in diameter batter balls with help of small ice cream scoop. You will have between 35-40 balls.
To Make Icing: line a sieve with a clean bowl. Sift 2 cups powdered sugar. Add pinch of salt, 1/4 cup milk and 2 drops vanilla extract. Whisk ingredients until smooth.
Drop 10 or less balls at a time into oil. These will double in size right away. When one side is cooked, turn to the other if they didn't turn by themselves. Heat each side until golden brown.
Transfer cooked doughnuts onto tray lined with paper towel to absorb any excess oil and to slightly cool (30 seconds). Roll doughnuts in prepared icing and transfer to a serving tray. Continue until all doughnut holes are fried.