Make the nut crust: Place the butter for the nut crust in a medium saucepan over medium-high heat. Cook for 4 minutes, until the butter is nutty smelling and golden brown. Remove from the heat and strain the butter through a fine-mesh sieve to remove any black bits. Add the pistachios, pine nuts, sugar, lemon juice, and a dash of salt. Mix well, then spread out in a small parchment-lined baking sheet, about 6 inches wide and 8 inches long. Chill in the fridge for 2 to 3 hours (or if in a rush – the freezer for 30 minutes) until the butter has set firmly, then cut the mixture into 2 equal rectangles. Return the rectangles to the fridge until ready to use.
To make the vichyssoise: bring a medium saucepan of salted water to a boil and add the parsley and arugula leaves. Blanch for 30 seconds, then refresh under cold water. Strain, squeeze out the excess water, set aside to dry, then coarsely chop. Place 1 tablespoon of the oil and the butter in a medium saucepan over medium heat. Add the shallot and sauté for 3 to 4 minutes, stirring every now and then, until soft but not colored. Add the garlic and leek and cook for another 2 to 3 minutes. Add the potato and cook for 5 to 6 minutes, stirring frequently, until shiny and glossy. Pour over the chicken stock and bring to a boil over medium-high heat, then simmer for 8 to 10 minutes, until cooked but still retaining a bite. Add the blanched parsley and arugula and cook for a final minute, then remove from the heat. Add the spinach, along with a teaspoon of salt and a good grind of black pepper. Transfer to a blender, blitz well until completely smooth, and set aside until ready to use.
Prepare the fish: Preheat the broiler to 400° F, or to its highest setting. Spread the sea bass fillets out on a parchment-lined (broiler-safe) baking pan and brush each with the olive oil. Season with 1/2- to 1-teaspoon of salt and a good grind of black pepper and cook for 8 to 9 minutes, until the sea bass is almost cooked. Remove the baking pan from under the broiler and lay a rectangle of nut butter on top of each fillet. Return to the broiler and cook for a final 2 minutes, until the crust is golden brown. Remove from the oven and squeeze over the lemon juice. Warm the vichyssoise and spoon it into shallow wide bowls. Lay a sea bass fillet on top, and serve at once.