In a large mixing bowl, combine the eggs, sugar, vanilla, vegetable oil, buttermilk, flour, baking powder, baking soda, and salt. The dough will be quite sticky.
Roll out the dough on a floured board. Cut into rounds with a donut cutter.
Heat a pot of oil to a depth of at least two inches, and fry the donuts, flipping once, until golden brown and cooked through. Drain on a rack set over paper toweling.
Cooled donuts can be packed in an airtight container and frozen for up to 6 weeks.
Notes
Store donuts in an airtight container and enjoy within 3 days. For longer storage, freeze the donuts in a freezer-safe container.