Preheat your oven to 350°F and generously spray a donut pan with non-stick cooking spray.
In a large bowl, whisk together the flour, sugar, baking powder, pumpkin pie spice, and salt.
In another bowl, mix the pumpkin puree, milk, eggs, melted butter, and vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and stir just until combined—don't overmix or your donuts will be tough!
Spoon or pipe the batter into the prepared donut pan, filling each cavity about 2/3 full.
Bake for 10-12 minutes, until a toothpick inserted comes out clean. Let cool in the pan for 5 minutes, then transfer to a wire rack.
While the donuts cool, make the glaze by whisking together the powdered sugar, maple syrup, and milk until smooth and pourable.
Dip the top of each donut into the glaze, letting the excess drip off. Place back on the wire rack and let the glaze set for about 10 minutes before serving (if you can wait that long!).