Heat the oven to 350°F (175°C). Brush a 12-cavity donut pan generously with melted butter or nonstick spray, making sure every cavity is coated.
In a large bowl, whisk the flour, sugar, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger until evenly mixed, with no lumps of spice.
In a second bowl, whisk the pumpkin purée, milk, oil, egg, and vanilla until smooth. Stir into the dry ingredients until just combined, then spoon the batter into the cavities, about three-quarters full.
Bake for 12 to 15 minutes, checking at 12. The donuts are done when a toothpick inserted in the center comes out clean and the tops spring back when lightly pressed. Cool in the pan for 5 minutes.
Melt 4 tablespoons of butter. Stir the 1/2 cup of sugar with 1 teaspoon of cinnamon in a shallow dish. Dip each warm donut in the butter, then roll it in the cinnamon sugar. Let them cool on a wire rack.