Preheat your oven to 350°F (175°C) and grease and flour a 9x13 inch baking pan.

Sift together all-purpose flour, granulated sugar, brown sugar, baking powder, salt, cinnamon, and nutmeg in a large bowl.

In another bowl, whisk together melted butter, vegetable oil, eggs, and vanilla extract until smooth.

Pour the wet mixture into the dry ingredients, stirring gently until just combined.

Fold in the peeled and chopped apples and the chopped nuts if using, ensuring even distribution.

Pour the batter into the greased pan and smooth the top.

Bake for 45 to 50 minutes, checking doneness with a toothpick.

Let the cake cool in the pan for about 10 minutes before transferring to a wire rack.

Dust with powdered sugar if desired, cut into squares, and serve warm or at room temperature.