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 1 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, 40 to 45 minutes. Cool in pan 15 minutes. Run a knife around edge of cake to loosen, then invert onto a plate. Reinvert cake onto a rack to cool completely.