In a large mixing bowl combine the pumpkin, brown sugar, granulated sugar, butter, applesauce, and vanilla. Mix until incorporated.
Add the eggs to the bowl and mix well.
In a separate bowl whisk together the flour, pumpkin spice, cinnamon, baking powder, baking soda, and salt.
Add the dry ingredient mixture to the wet ingredients and mix until fully blended, scraping the sides as needed.
Pour the batter into the prepared baking dish and bake for 32-36 minutes or until a toothpick inserted comes out clean or the center springs back when gently pressed.
Allow the cake to cool for 60 minutes on a cooling rack before adding the frosting.
Frosting
In a large bowl add the cream cheese, butter, and vanilla and beat until creamy.
Gradually add the confectioner's sugar, milk, pumpkin pie spice, and salt to the bowl. Mix slowly for 20 seconds and then increase to medium-high speed and beat for 2 minutes until the frosting is thick and fluffy while scraping the sides as needed.
Add the frosting to the top of the cake, spreading into a thick, even layer.
Place the walnuts in a zip-top bag and roll with a rolling pin to crush. Or you can chop them on a cutting board - whichever works best for you.
Sprinkle the crushed walnuts on top of the frosted cake.
Serve immediately or within 3 days.
Notes
Heavy cream or almond milk can be used in substitution for the milk.
If you would like more spice you can add ½ teaspoon to 1 teaspoon more cinnamon or pumpkin spice.
The cake can be stored in a sealed container for up to 5 days.