Preheat oven to 325 degrees and lightly spray or brush a doughnut pan with cooking oil.
In a large bowl, whisk together flour, sugar, baking powder, baking soda, and salt.
In a separate smaller bowl, lightly whisk eggs. Add in buttermilk, melted butter, and vanilla extract, and whisk until combined.
Stir the wet ingredients into the dry ingredients until just combined, taking care not to over-mix (over-mixing can result in tough donuts).
Fill each cavity in the doughnut pan about ¾ of the way with batter. I like to put batter in a plastic bag, cut off the corner, and squeeze the batter into the doughnut pan (you can also use a pastry bag).
Bake donuts for 10-13 minutes, until they're light and springy to the touch. Let donuts cool in the pan for a few minutes and then remove to cool completely on wire rack. If you're using a 6-cavity donut pan, repeat with remaining batter.
Once doughnuts are completely cooled, dip upside-down in icing to coat. Set on wire rack with waxed or parchment paper set underneath and top with sprinkles, if desired.