Preheat your oven to 350°F (175°C). Grease a donut pan lightly with butter or non-stick spray.
In a large bowl, whisk together flour, sugar, baking powder, baking soda, salt.
In another bowl, whisk together buttermilk, eggs, melted butter, and vanilla extract.
Pour the wet ingredients into the dry ingredients and stir until just combined. Avoid overmixing.
Spoon or pipe the batter into the donut pan cavities, filling each about 3/4 full.
Bake for 10-12 minutes, or until the donuts are lightly golden and a toothpick inserted comes out clean.
Let the donuts cool in the pan for 5 minutes, then transfer to a wire rack to cool completely.
For the pumpkin frosting, in a bowl, whisk together powdered sugar, pumpkin puree, milk, cinnamon, and nutmeg until smooth. Adjust milk to get a thick but dip-friendly consistency.
For the classic glaze, mix powdered sugar, vanilla, and enough milk to make a thin drizzle consistency.
Once the donuts are cooled, dip the top of each into the pumpkin glaze, letting excess drip off.
Using a spoon or piping bag, drizzle the classic icing over the glazed donuts in a decorative pattern.
Allow the glaze and drizzle to set for about 15-20 minutes before serving.