Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan or line it with parchment paper.
In a large mixing bowl, combine the flour, granulated sugar, baking powder, baking soda, ground cinnamon, and salt. Whisk together until well combined.
In another bowl, mix the softened butter and applesauce until smooth.
Add the eggs and vanilla extract to the butter and applesauce mixture. Beat until well blended.
Gradually add the wet ingredients into the dry ingredients, mixing until just combined. Be careful not to overmix.
Fold in the diced apples and chopped nuts (if using). The batter should be thick and chunky with apple pieces.
Pour the batter into the prepared baking pan and spread evenly.
Bake in the preheated oven for 35-40 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, allow the cake to cool in the pan for about 15 minutes before transferring it to a wire rack to cool completely.
If desired, dust the top of the cooled cake with powdered sugar before serving.
Notes
Slice the cake into squares and garnish with powdered sugar and whipped cream or vanilla ice cream.