Preheat the oven to 350 degrees F. Lightly grease and flour a 9-inch round cake pan and set aside.
Combine the eggs and sugar in mixer bowl and beat on medium speed until light and fluffy. Mix in the olive oil, applesauce, and vanilla.
Combine the flours, hazelnuts, baking powder, and salt. Using low speed, add gradually to the mixer bowl alternately with the almond milk to make a well mixed batter. Fold in the shaved chocolate by hand, then pour the batter into the prepared pan.
Bake for 20-25 minutes, until a toothpick inserted in the center comes out clean. Remove to a cooling rack. When cake is cool, run a table knife around the edge, then turn the cake out onto a serving plate.
Mix the confectioner’s sugar with the cocoa powder. Stir in the coconut oil and vanilla. Slowly stir in the hot water until a pourable consistency is reached. Drizzle over the cake in a decorative pattern, then sprinkle with the chopped hazelnuts.