Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a medium bowl, whisk together the whole wheat flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt.
In a large bowl, beat the eggs until frothy. Add the unsweetened applesauce, plain Greek yogurt, melted coconut oil, and vanilla extract. Mix until well combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined.
Fold in the diced apples and chopped walnuts (if using) until evenly distributed.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center 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 serving.