Prepare the oven. Preheat your oven to 350°F and spray a 9x13 lipped baking sheet with nonstick baking spray.
Combine the dry ingredients. In a large bowl, combine the all-purpose flour, cinnamon, baking soda, ground nutmeg, salt, and allspice and set aside.
Mix the wet ingredients. In a medium bowl, mix together the applesauce, brown sugar, granulated sugar, butter, eggs, and vanilla extract. It's ok if the butter doesn't completely blend in.
Add the wet to the dry. Add the wet ingredients to the bowl with the dry ingredients and stir together until smooth, but do not overmix.
Toss in walnuts for crunch. Add the walnuts to the mixture.
Transfer to your baking dish. Pour the batter into the baking dish and bake uncovered for 30 minutes or until a toothpick inserted into the center of the cake comes out clean.
Cool and serve. Cool completely before slicing and serving.
Optional Glaze
While waiting for your cake to cool, heat 2 tablespoons of butter and 1 tablespoon dark brown sugar in a small microwave-safe bowl for 1 minute. Stir until the brown sugar has dissolved and brush over the cake before slicing.