In a large mixing bowl, whisk together the eggs, sourdough starter, milk, pumpkin puree, sugar, melted butter, and vanilla until smooth.

In a separate bowl, whisk together the flour, pumpkin pie spice, baking powder, baking soda, and salt.

Add the dry ingredients to the wet ingredients. Stir just until the flour is incorporated and the batter is smooth. Avoid overmixing.

Preheat the oven to 350°F. Lightly grease a donut pan with nonstick cooking spray or softened butter.

Transfer the batter to a piping bag or zip top bag with one corner cut off. Pipe the batter into the donut cavities, filling each approximately three quarters full.

Bake for 10 to 11 minutes, or until the tops spring back when lightly pressed and a toothpick comes out clean.

Allow the donuts to cool in the pan for about 5 minutes. Carefully remove them and transfer them to a wire rack to cool completely.

When ready to bake, preheat the oven to 350°F. Stir the batter gently, transfer it to the prepared donut pan, and bake as directed. The cold batter may require an additional 1 to 2 minutes of baking time.

Whisk together the melted butter, maple syrup, powdered sugar, and vanilla until smooth.

Add milk, ½ teaspoon at a time, if the glaze is too thick for dipping.

Dip the tops of the completely cooled donuts into the maple glaze. Place them on a wire rack and allow the glaze to set before serving.