Begin by melting butter and then allowing it to cool. Set to the side.
Warm milk on the stove or microwave until it is above room temperature but not hot.
In the bowl of your Kitchenaid mixer, add warm milk and stir in sugar. Sprinkle the yeast packet over top and allow to sit for five minutes. The yeast should begin to look frothy.
Gently stir in sourdough starter, eggs, melted butter, and salt.
In stages, add in flour and stir using the dough hook attachment on the Kitchenaid. Knead the dough on a low speed for about 5 minutes until the dough is neither dry nor overly sticky. This recipe creates a pretty smooth dough. (Note: This can be done by hand, it will just take a little bit of elbow grease. Knead by hand as best you can)
Place dough into a bowl and place in the fridge overnight. (If you don't have the time, wait at least until the dough is chilled.
Remove the chilled dough from the fridge and roll to ½-inch thickness. Cut into donut shapes. (Note: I used a glass for the main part and a cocktail shaker lid for the middle/donut hole. Needless to say, you do not need a specific donut shape cutter. Find something you have on hand in the kitchen).
Place donuts onto parchment paper lined baking sheets and allow to rise at room temperature for about an hour.
In a dutch oven, heat your oil to 350 degrees. While the oil is heating up, place paper towels on to a plate for you to place cooked donuts on to catch some of the excess oil.
Once the oil is hot, fry donuts. They cook quickly, about 1 minute per side. After one side becomes golden brown, flip and cook the second side the same way. Do not overcrowd with donuts, about 3 per batch is my go to. The donut holes cook more quickly, watch for color rather than time on these.
Carefully scoop donuts out with a slotted spoon and place on paper towels to drain. Once the excess oil has been absorbed, place the hot donuts on to a cooling rack and sprinkle liberally with powdered sugar. Serve warm.