Preheat your oven to 350°F (175°C).
Grease and flour a 9x13 inch baking pan.
In a large bowl, combine the flour, granulated sugar, brown sugar, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg.
In another bowl, whisk together the melted butter, eggs, and vanilla extract.
Gradually add the butter mixture to the dry ingredients, stirring until just combined.
Fold in the chopped apples and walnuts.
Pour the batter into the prepared baking pan and spread it evenly.
Bake in the preheated oven for 45-50 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan on a wire rack before slicing and serving.