Preheat your oven to 350F (175C).
If needed, chop and lightly toast the hazelnuts.
Whisk together eggs and brown sugar until light and fluffy, about 5 minutes. You can do this by hand but it's much easier to do with a hand mixer or stand mixer!
Add vanilla extract and mix.
Mix together dry ingredients in a separate bowl - flour, baking powder, and salt.
Mix together wet ingredients (coffee/espresso and vegetable oil) in a jug or bowl.
Alternate adding the dry and wet ingredients into the egg mixture. You should fold the ingredients in carefully with a spatula if doing this by hand. If using a mixer, make sure to run the mixer until the ingredients are just combined, maybe around 10 seconds for each addition.
Add in the chopped hazelnuts. and mix until evenly distributed in the batter.
Line a round, 10 inch (or about 25cm) diameter springform pan with parchment paper. Make sure to grease the edges.
Pour the batter into the pan.
Bake for about 20-25 minutes, or until a knife inserted into the center comes out clean. Make sure to check right at 20 minutes and you can always leave it in the oven for longer if it's not ready.
Remove from the pan and let cool.
To make the icing, mix together powdered sugar and espresso until you have a thick icing. Add (the tiniest!) splash of water if it's not coming together.
Pour over the top of the cake and spread to the edges. Decorate with chopped hazelnuts and enjoy!