First, preheat your oven to 175°C (350°F) and grease or line an 8-inch cake pan with parchment paper to prevent sticking and ensure easy removal after baking.
Next, spread the hazelnuts onto a baking tray and toast them for 8–10 minutes until fragrant and lightly golden. Allow them to cool completely before chopping them into small pieces for both the cake and crumble.
Meanwhile, combine the flour, brown sugar, granulated sugar, melted butter, chopped toasted hazelnuts, salt, and cinnamon, if using, in a bowl. Mix everything together until the mixture resembles coarse crumbs with a few larger clusters. Place the crumble in the refrigerator while preparing the cake batter to keep it crisp during baking.
In another bowl, whisk together the flour, baking powder, baking soda, and salt until evenly combined.
In a large mixing bowl, whisk together the melted butter, granulated sugar, dark brown sugar, and eggs until smooth and well combined. Then add the cooled coffee, buttermilk, Greek yogurt, and vanilla extract, whisking until the mixture becomes creamy.
Gradually fold the dry ingredients into the wet ingredients using a spatula. Mix only until no dry flour remains to keep the cake light and tender. Finally, fold in the chopped toasted hazelnuts until evenly distributed throughout the batter.
Pour the batter into the prepared cake pan and spread it into an even layer. Sprinkle the chilled hazelnut crumble generously over the top, making sure it evenly covers the surface.
Bake the cake for 35–45 minutes, or until the crumble is beautifully golden and a toothpick inserted into the center comes out clean or with a few moist crumbs attached.
While the cake is baking, combine the dark brown sugar, granulated sugar, coffee, butter, salt, and vanilla, if using, in a small saucepan. Cook over medium heat, stirring continuously until the butter melts and the drizzle becomes smooth and slightly thickened.
Allow the cake to cool for about 10–15 minutes before drizzling the warm coffee sauce over the top. Slice and serve slightly warm or completely cooled with a fresh cup of coffee or tea.