Spread the hazelnuts on a baking sheet and then place in the oven. Toast for 8-10 minutes or until lightly browned, stirring once halfway through. Remove from the oven and allow to cool slightly. Once cool enough to touch, coarsely chop the nuts and then toss them with 1 tsp hazelnut oil and a hefty pinch of salt.
Meanwhile, stir together the diced shallot, pomegranate juice, sherry vinegar, rice vinegar, and ½ tsp salt in a medium bowl. Allow to rest for 5 minutes. Whisk in the olive oil and the remaining 1 tbsp of hazelnut oil. Season to taste with salt, black pepper, and extra vinegar.
Combine the persimmons, sliced shallots, and pomegranate seeds in a large salad bowl. Add in the dressing and toss to combine. Season again with salt and black pepper, to taste. Toss with the arugula and taste again for seasoning.
Arrange the salad among 4 bowls and sprinkle the hazelnuts over the top.