Preheat the oven to 350 degrees F. Grease a 9 x 13 baking dish with butter and set aside.
In a medium bowl, whisk the flour, baking soda, baking powder, pumpkin spice, cinnamon, and salt until thoroughly combined.
In a large bowl, add the ½ cup butter, brown sugar, white sugar, pumpkin puree, applesauce, and vanilla extract. Mix with a hand mixer until incorporated.
Add the eggs to the bowl and mix well.
Add the dry ingredients 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 32-35 minutes or until a toothpick inserted comes out clean.
Allow the cake to cool on a cooling rack for at least 60 minutes before frosting. Do not frost a warm cake; the frosting will melt.
Spread the frosting evenly with a silicone spatula and top with the crushed walnuts or pecans.
Cream Cheese Frosting
In a large bowl, add the cream cheese, butter, and vanilla and mix until creamy.
Gradually add the powdered sugar, pumpkin pie spice, milk, and salt to the bowl. Mix slowly for 20 seconds, then increase to medium to high speed and mix for 2 minutes until the frosting is thick and fluffy while scraping the sides as needed.
Notes
Hint: Place the walnuts in a ziplock bag and roll with a rolling pin to create a finer consistency. You can also use a mortar and pestle.
The cake can be stored in a sealed container for 3 days.
The cake can be frozen unfrosted for up to 3 months. Wrap it snugly in plastic wrap first. Allow it to defrost in the refrigerator.
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 of cinnamon or pumpkin spice.