Preheat your oven to 350° ℉
Butter and flour, very well, a 9" x 3"tube pan.
In the bowl of a stand mixer, or using a large bowl with a hand mixer, place the sugar, eggs, egg yolks and salt and beat for 4-5 minutes on high speed with the whisk attachment. The mixture should be light in color and tripled in volume.
Add the vegetable oil and whisk on medium speed until incorporated.
Whisk together the flour and the baking powder.
Sift half the dry ingredients into the bowl and beat on low speed just until incorporated.
Add the buttermilk and continue to whisk on low speed until incorporated.
Sift in th remaining dry ingredients and whisk until combined.
Whisk together the instant espresso powder and water until you have a smooth paste.
Divide the dough in half. Add the espresso paste to one half and stir to combine. Add the ground hazelnuts to the other half and stir to combine
Randomly add the 2 batters to the prepared pan. and swirl with a flat knife. Pour the batter into the prepared pan and bake for about 30 minutes, or until the top of the cake springs back when gently pressed. Do not over bake or the cake will be dry.
Let cool in the pan for about 10 minutes.
Remove from the pan and place on a rack to cool completely