1. 1. Combine Wet Ingredients: In a large mixing bowl, whisk together the active sourdough starter, pumpkin puree, granulated sugar, egg, and vanilla extract until well combined and smooth. The mixture will be a lovely orange color.

2. 2. Mix Dry Ingredients: In a separate medium bowl, whisk together the all-purpose flour, baking powder, pumpkin pie spice, and salt. Make sure there are no lumps in the dry ingredients.

3. 3. Form the Dough: Gradually add the dry ingredients to the wet ingredients, alternating with the milk, mixing until just combined. Be careful not to overmix; a few streaks of flour are fine. The dough will be soft and slightly sticky.

4. 4. Roll and Cut Donuts: Lightly flour a clean surface. Turn the dough out onto the floured surface and gently roll it to about 1/2-inch thickness. Use a donut cutter or two different-sized round cookie cutters (one larger for the donut, one smaller for the hole) to cut out the donut shapes. You can re-roll scraps once or twice to get more donuts.

5. 5. Heat Oil for Frying: Pour vegetable oil into a large, heavy-bottomed pot or Dutch oven to a depth of about 2-3 inches. Heat the oil over medium heat until it reaches 350 F (175 C). Use a candy thermometer to monitor the temperature; maintaining the correct temperature is crucial for perfectly cooked donuts.

6. 6. Fry the Donuts: Carefully place 2-3 donuts into the hot oil, being careful not to overcrowd the pot. Fry for 1-2 minutes per side, or until golden brown and cooked through. The donuts will puff up beautifully.

7. 7. Drain and Glaze: Use a slotted spoon or spider to remove the fried donuts from the oil, allowing excess oil to drip off. Transfer them to a wire rack lined with paper towels to drain. While the donuts are still warm, whisk together the powdered sugar, milk, and vanilla extract in a shallow bowl to make the glaze. Dip each donut into the glaze, turning to coat both sides, then return to the wire rack to set.