Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan with gluten-free flour.
In a medium bowl, whisk together the gluten-free flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, whisk the coconut sugar, maple syrup, and vegetable oil until smooth.
Add the eggs one at a time to the wet mixture, whisking well after each addition.
Stir in the vanilla extract and unsweetened applesauce until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, folding gently until just combined.
Fold in the diced apples and chopped walnuts (if using) evenly into the batter.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 15 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.