Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x9-inch baking dish.

Beat sugar, melted butter, and egg in a large bowl with an electric mixer until fluffy. Fold in the apples and walnuts. Sift flour, salt, baking powder, baking soda, and cinnamon together in a separate bowl; stir into apple mixture until just blended. Spread batter evenly in the prepared baking dish.

Bake in the preheated oven until a toothpick inserted in the center comes out clean, about 35 minutes.