Preheat oven to 350°F with rack in middle. Butter a 8 or 9 inch square baking dish.
Whisk together flour, baking powder, baking soda, salt, and spices in a medium bowl.
In large bowl beat butter, brown sugar, and vanilla with an electric mixer until pale and fluffy, 2 to 3 minutes. Beat in eggs and applesauce. At low speed, mix in flour mixture until just combined, then stir in walnuts.
Pour into greased baking dish and bake until golden-brown and a wooden pick inserted into center comes out clean, 40 to 45 minutes. COOL COMPLETELY.
Combine all frosting ingredients together (cream cheese, butter, powdered sugar,cinnamon and vanilla) with an electric mixer until smooth. You may need to add a little milk to make it spreadable if your butter wasn’t soft enough.
Spread frosting evenly on top of cake. Cut the cake into any size you like. Decorate with pecans.