Add the sourdough starter, bread flour, milk, eggs, salt, vanilla extract, brown sugar, pumpkin puree, cinnamon, and ginger to the bowl of a stand mixer fitted with a dough hook.

Mix on low speed until the ingredients come together into a shaggy dough.

With the mixer running on low, add the softened butter about one tablespoon at a time. Allow each addition to mostly incorporate into the dough before adding the next.

The dough may remain slightly soft and tacky because of the pumpkin, but it should have enough gluten development to stretch without immediately tearing.

Transfer the dough to a large, lightly greased bowl and cover tightly.

Look for the dough to increase in volume by approximately 50–60%. Because this is an enriched dough containing eggs, butter, sugar, and pumpkin, it may ferment more slowly than a traditional sourdough bread dough.

Once bulk fermentation is complete, you can shape the donuts immediately or tightly cover the dough and refrigerate it overnight.

Turn the dough out onto a lightly floured work surface.

Roll the dough to approximately 1/2-inch thickness.

Using a donut cutter, cut out the donuts and donut holes. Gently gather and reroll the scraps as needed, being careful not to overwork the dough.

Place each donut onto an individual square of parchment paper and arrange them on a baking sheet.

Loosely cover the donuts and allow them to rise in a warm place for approximately 2–3 hours, or until visibly puffy and light.

To check the fermentation, gently press your finger into the side of a donut. If the indentation slowly springs back, the donuts are ready to fry. If the dough immediately springs back, allow them to continue rising.

Avoid letting the donuts dramatically double in size. Properly proofed donuts should look airy and puffy while still being sturdy enough to handle.

Add coconut oil or another neutral frying oil to a heavy-bottomed pot or Dutch oven until the oil is approximately 2–3 inches deep.

Heat the oil to 340–350°F (171–177°C).

Carefully lower the donuts into the oil using their individual parchment squares. Peel the parchment away once the donut releases into the oil.

Fry only a few donuts at a time so the oil temperature does not drop significantly.

Fry for approximately 1–2 minutes per side, or until the donuts are evenly golden brown and cooked through.

Keep a close eye on the oil temperature throughout frying. Pumpkin and brown sugar can cause these donuts to brown relatively quickly, so keeping the oil near the lower end of the frying range can help prevent the outsides from becoming too dark before the centers finish cooking.

Add the powdered sugar, maple syrup, milk, and vanilla extract to a medium bowl.

Whisk until completely smooth.

For a thinner glaze, add a small splash of additional milk. For a thicker glaze, whisk in additional powdered sugar.

Allow the freshly fried donuts to cool for several minutes so they are warm rather than piping hot.

Dip the tops of the donuts into the maple glaze, allowing any excess glaze to drip back into the bowl.

Return the donuts to the wire rack and allow the glaze to set before serving.

For the best texture, enjoy the donuts the same day they are fried.