Preheat over to 325F. Grease a 9-inch springform pan with a little oil and line the base with parchment paper.
In a bowl, measure and sift the cocoa powder. Whisk in the hot water until you have a smooth, chocolatey, still-runny paste. Whisk in the vanilla extract, then set aside to cool a little.
In a small bowl, combine the ground hazelnuts and baking powder. In a large bowl, add the sugar, ½ cup olive oil, and eggs, and whisk together until you have a pale, thickened cream. Or use a stand mixer with the paddle attachment, and beat together vigorously for about 3 minutes until you have a pale, thickened cream.
Add the cocoa mixture to the egg mixture, and whisk until well combined. Add the ground hazelnut mixture and combine. If using a mixer reduce the speed a little and pour in the cocoa mixture, beating as you go. When all is scraped down in the mixer, slowly add the ground hazelnut mixture.
Scrape down, stir a little with a spatula, then pour the batter into the prepared pan. Bake for 40-45 minutes or until the sides are set and the very center on top still looks slightly moist. A cake tester should come out with a few sticky crumbs attached.
Let the cake cool for 10 minutes on a wire rack, still in its pan. Then, using a small metal spatula, loosen the sides of the cake from the pan. Allow to cool completely, or serve while still warm. Dust the powered sugar over the top. Serve with a scoop of vanilla ice cream, vanilla gelato, or mascarpone on the side with crushed amaretti on top of the ice cream, gelato, or mascarpone.