Preheat oven to 400°F. Lightly season the black cod fillets with kosher salt and black pepper. Lightly dust the fillets in flour, shaking off excess. In a large cast iron skillet, over medium-high heat, preheat the vegetable oil until just smoking. Sear the fillets on the side that was connected to the bone until lightly browned, about five minutes. Using a wide stainless steel spatula, turn the fillets over, and transfer to oven to cook for about 5 minutes more. In a large mixing bowl, toss the 4 fresh herbs. To serve, divide the herb salad equally among four large, warm plates.
Brown Butter Sauce
In a small saucepan, over medium heat, melt and brown the butter, about 3 to 4 minutes. Remove the pan from the heat, add the shallots and lemon juice and correct the seasoning to taste with kosher salt and black pepper. Place a fillet on each plate next to the herb salad. Splash the brown butter sauce over the salad and fillet. Serve immediately.