Whisk together flour, baking powder, baking soda, salt, and spices.
Beat butter, brown sugar, and vanilla with an electric mixer at high speed until pale and fluffy, 2 to 3 minutes. Add eggs 1 at a time, beating well after each addition, then beat in applesauce. At low speed, mix in flour mixture until just combined.
Spread batter evenly in pan and bake until golden-brown and a wooden pick inserted into center comes out clean, 40 to 45 minutes. Cool in pan 15 minutes. Remove cake from the pan or spring form (you could leave it in and cut from the pan if you wish).
Beat cream cheese, butter, and vanilla with an electric mixer at high speed until fluffy. Sift confectioners sugar and cinnamon over cream cheese mixture, then beat at medium speed until incorporated.
Spread frosting over top of cooled cake. Top with walnuts.