A true delight.
Mix starter, eggs, milk, melted butter, vanilla, and sugar in a large bowl. Stir together flour, salt and baking powder and add to the starter mixture; mix thoroughly (dough should be the consistency of a soft biscuit dough). Roll dough out on a floured surface to 1/2-inch thickness and cut with a well-floured 3" to 4" doughnut cutter (or use a 3" to 4" biscuit cutter to cut rounds, then use a small thimble-like cap to cut out the holes.)
Meanwhile, heat oil to a depth of about 3" to 4" in a large, deep-sided pan to 375° This can take 15 to 20 minutes over medium heat. Gently lower doughnuts into oil, frying only 3 or 4 at a time. Don't crown pan. Cook until golden brown on both sides, turning once, 2 to 3 minutes total for doughnuts and about a minute for holes. *The doughnuts will expand while frying, so try not to crowd pan.*
When done, remove with slotted spoon and drain on racks set over paper towels. If desired, roll warm doughnuts in a mixture of 1 cup water and 2 teaspoons ground cinnamon. Serve immediately.