Preheat the oven to 350 degrees F and grease a 6-inch round cake pan.
In a mixing bowl, whisk together the flour, sugar, cocoa powder, baking powder, baking soda, and salt.
Add the chocolate hazelnut spread, milk, oil, egg, and vanilla extract. Stir until the batter is smooth and no dry streaks remain.
Pour the batter into the prepared pan and bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean. Let the cake cool completely in the pan.
Melt the chocolate chips in a microwave-safe bowl in 30-second bursts, stirring between each until smooth.
Place the cooled cake on a plate. Spread a thin layer of melted chocolate over the top. Set one sugar cone upright in the center to form the hat point. Spoon more melted chocolate over the cone and the cake edges to create the brim and seal everything together.
Sprinkle chopped hazelnuts around the base while the chocolate is still soft. Let the cake sit until the chocolate sets.