Preheat oven to 350°F with rack in middle. Butter an 8- or 9-inch square cake pan.

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 one at a time, beating well after each addition, then beat in applesauce.

At low speed, mix in flour mixture until just combined, then stir in walnuts (if using).

Spread batter evenly in pan and bake until golden-brown and a wooden pick inserted into center comes out clean, about 35 to 40 minutes.

Cool in pan for 15 minutes. Run a knife around edge of cake to loosen, then invert onto a plate. Reinvert cake onto a rack to cool completely.

Make frosting: 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.