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, ground cinnamon, ground nutmeg, and salt.
In a large bowl, combine the grated apples, unsweetened applesauce, plain whole milk yogurt, vegetable oil, egg, and vanilla extract. Mix well until fully combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared cake pan and spread evenly.
Bake in the preheated oven for 30 to 35 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 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.