Preheat oven to 350°F/175° C. Position racks in upper and lower thirds of the oven.
Whisk together flour, cocoa, baking soda, and salt in a bowl until combined. Stir together buttermilk and vanilla in a measuring cup.
In the bowl of the stand mixer, beat together butter and brown sugar at medium-high speed until pale and fluffy, about 3 minutes (or 5 minutes if using hand mixer), then add egg, beating until combined well. Reduce speed to low and add 1/3 of the flour mixture. Mix and scrape down the sides of the bowl, then add 1/2 the buttermilk and mix until smooth. Continue with another 1/3 of the flour, then the rest of the buttermilk, and ending with the rest of the flour mixture, scraping between each addition. The final batter should be thick and smooth.
Scoop batter onto prepared baking sheets, about 2 inches apart.
Bake in preheated oven, switching position of sheets halfway through baking, until tops are puffed and cakes spring back when touched, 11 to 13 minutes. Remove from oven and allow the cookies to rest on the baking sheet for a couple minutes, then transfer with a metal spatula to a rack to cool completely, at least a couple hours.
Spread a rounded tablespoon of Nutella on flat sides of half the cakes. Sprinkle some of the chopped hazelnuts over the Nutella, then top with remaining cakes.