In a large bowl, whisk together the flour, baking powder, baking soda, pumpkin pie spice, cinnamon, and salt. Set aside.
In another bowl, whisk the sugar and eggs until smooth and slightly thickened. Add the pumpkin puree, sour cream, melted butter, and vanilla extract, then mix until fully combined.
Gradually add the dry ingredients to the pumpkin mixture, stirring gently until a soft dough forms. Avoid overmixing to keep the donuts tender.
Cover the dough and refrigerate for at least 30 minutes. Chilling helps the dough become easier to handle and creates a better old-fashioned donut texture.
Lightly flour a work surface and roll the dough to about ½-inch thickness. Cut out donuts using a donut cutter or two round cutters, then transfer them to a lined baking sheet.
Heat the oil in a deep fryer or heavy-bottomed pot to 350°F (175°C).
Fry the donuts in small batches for about 1 to 2 minutes per side, until golden brown and cooked through. Avoid overcrowding the oil, as it can lower the temperature and make the donuts greasy.
Transfer the donuts to a wire rack lined with paper towels to drain excess oil.
For the glaze, whisk together powdered sugar, milk, vanilla extract, and salt until smooth. Adjust the consistency with a little more milk if needed.
Dip the slightly warm donuts into the glaze, allowing the excess to drip off. Place them back on the rack and let the glaze set before serving.