Add the active sourdough starter, milk, flour, eggs, salt, maple syrup, and vanilla extract to the bowl of a stand mixer fitted with the dough hook attachment.

Mix on low speed until the ingredients are combined and a shaggy dough forms.

Add the softened butter, one tablespoon at a time, with the mixer running on low. Allow each addition to become fully incorporated before adding the next.

Once all of the butter has been incorporated, increase the mixer speed to high and knead for about 7–10 minutes, or until the dough becomes smooth, elastic, and begins pulling away from the sides of the bowl.

Transfer the dough to a lightly greased bowl and cover tightly with plastic wrap.

Allow the dough to bulk ferment at room temperature until it has risen by approximately 50–60%.

Cut 9 parchment squares that will fit the donuts.

Turn the dough onto a lightly floured work surface and divide it into 9 equal pieces, approximately 100 g each.

Working with one piece at a time, roll it into a rope approximately 12 inches (30 cm) long.

Fold the rope in half to create an upside-down U shape. Twist the two strands around each other several times, then pinch the ends firmly together at the bottom to secure the twist.

Place each shaped donut on its own square of parchment paper.

Cover the shaped donuts loosely with greased plastic wrap.

Allow them to proof at room temperature for approximately 1–3 hours, or until they look noticeably puffy and airy.

Whisk together the powdered sugar, maple syrup, milk, and vanilla extract until smooth.

Pour 2–3 inches of neutral oil into a heavy-bottomed Dutch oven or large pot. Heat the oil to 340–360°F (171–182°C).

Working in batches, carefully lower the donuts into the hot oil.

Fry the donuts for approximately 1–2 minutes per side, or until they are evenly deep golden brown.

Transfer the fried donuts to the prepared wire rack and allow the excess oil to drain.

While the donuts are still warm, dip the top of each donut twist into the maple glaze.

Allow any excess glaze to drip back into the bowl, then place the donut back on the wire rack to allow the glaze to drip and set.