Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan with gluten-free flour or line it with parchment paper.
In a medium bowl, whisk together the gluten-free flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg. Set aside.
In a large bowl, using an electric mixer, cream the softened butter, granulated sugar, and light brown sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Then mix in the vanilla extract.
Add half of the dry ingredients to the wet ingredients and mix on low speed until just combined.
Add the Greek yogurt and mix until incorporated.
Add the remaining dry ingredients and mix until just combined, being careful not to overmix.
Fold in the diced apples and chopped walnuts (if using) gently with a spatula.
Pour the batter into the prepared cake pan and smooth the top evenly.
Bake in the preheated oven for 45 to 50 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 serving.