Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking pan or line it with parchment paper.
In a medium bowl, whisk together the oat flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt until well combined.
In a large bowl, beat the eggs with the granulated sugar and brown sugar until the mixture is smooth and slightly fluffy.
Add the vegetable oil, vanilla extract, and Greek yogurt to the egg mixture and mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the diced apples and chopped walnuts (if using) until evenly distributed in the batter.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.