Preheat the oven to 350°F (175°C). Grease an 8-inch round cake pan and line the bottom with parchment paper.
In a large bowl, whisk the eggs and granulated sugar together until pale and fluffy, about 3-4 minutes.
Add the unsweetened applesauce and vanilla extract to the egg mixture and mix until combined.
In a separate bowl, combine the almond flour, baking powder, ground cinnamon, ground nutmeg, and salt.
Gradually fold the dry ingredients into the wet ingredients until just combined.
Toss the diced apples with the lemon juice to prevent browning, then gently fold them into the batter along with the chopped walnuts if using.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 45-50 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 before transferring it to a wire rack to cool completely.
Slice and serve the cake plain or with a dusting of powdered sugar if desired.