Preheat oven to 350F. Butter two 8-inch cake pans and line the bottom of the pans with a circle of parchment paper. Note: make sure your cake pans have full 2 inch sides to accomodate this cake batter.
In the bowl of a stand mixer or with electric beaters, cream the butter and sugar together until very light and fluffy (4-5 minutes).
Add the eggs and vanilla extract, and beat for another minute, scraping down the sides of the bowl to get everything incorporated.
Add the baking soda, baking powder, and salt, and mix briefly to combine.
Next, add the flour in a few stages, alternating with the half and half, mixing until all of the ingredients are combined. Finally fold in the ground hazelnuts. I like to finish mixing by hand with a silicone spatula to make sure everything on the bottom and the sides of the bowl is well incorporated.
Divide the batter evenly between the two prepared cake pans and bake for about 35-40 minutes, until risen, golden on top, and the cake springs back when pressed lightly.
Allow the cakes to cool for 10 minutes or so in the pan, then remove the layers and place on a baking rack to cool completely before frosting. (Cake layers can also be made a day ahead, and wrapped and kept in the fridge until ready to frost.)
For the frosting
Melt the chocolate in a double boiler or in the microwave using short bursts. Stir until smooth and allow to cool to room temperature before adding to the frosting. Note: this is important because you don't want warm chocolate to melt the butter.
Cream the butter and confectioner's sugar in the bowl of a stand mixer with the paddle attachment (or, if you have a food processor with a large enough bowl, you can use that, too).
Add the salt, the cooled melted chocolate, and the Nutella, and beat together until the mixture is smooth and uniform. Add the cream and mix until smooth. Note: adding cold cream can cause the frosting to split, so I like to have it at room temperature, if possible.
Frost between the cake layers, and then frost the sides and top of the cake. Pave the frosting along the sides to resemble bark. Use a fork to make a spiral pattern on the top of the cake to resemble the rings of a log.
Gently pat ground hazelnuts in patches around the side and bottom of the outside cake, if desired.
Dust the frosted cake with a little bit of confectioner's sugar to mimic freshly fallen snow. Place your meringue mushrooms on top and/or around the outside of the cake. Get ready for lots of oooohs and aaaahhhs!
Notes
To toast and grind hazelnuts:
Take 1 1/2 cups whole hazelnuts, skins on, and place in a single layer on a baking sheet. Toast in a 350F oven for 10-12 minutes, shaking the pan a couple of times during cooking, until fragrant. Let cool.
To grind, put the cooled nuts in a food processor and pulse until finely ground.
Measure out 1 1/4 cups for the cake, and reserve the rest for decorating.
To make this chocolate hazelnut cake ahead
Make the cake layers up to a day ahead. Let the layers cool completely, then wrap in plastic. You can also freeze the layers at this point, but be sure to double wrap well.
You can make the mushrooms a day ahead, but if you live in a humid climate, meringue can get soft. I like to keep the mushrooms uncovered, on the counter and they last just fine for me.
Make the frosting and frost the cake the day you want to serve.