Preheat the oven to 350°F (175°C).
In a large mixing bowl, combine the flour, sugar, baking powder, baking soda, pumpkin spice, and salt.
In another bowl, whisk together the pumpkin puree, eggs, and vanilla extract until well combined.
Gradually fold the wet ingredients into the dry ingredients until just combined.
Prepare a doughnut pan by greasing it lightly with non-stick spray.
Spoon the batter into the doughnut pan, filling each cavity about halfway.
Bake the doughnuts in the preheated oven for 10-12 minutes, or until a toothpick inserted comes out clean.
Allow the doughnuts to cool in the pan for 5 minutes before transferring them to a wire rack.
In the meantime, prepare the cream cheese filling by beating the softened cream cheese and powdered sugar until smooth.
If necessary, add a teaspoon of milk to adjust the consistency of the filling.
Once the doughnuts are completely cool, use a piping bag to fill them with the cream cheese mixture.
Serve the filled doughnuts immediately, or store them in an airtight container in the refrigerator.