In a blender or food processor, combine the shallot, anchovies, vinegar, lemon juice, half the Parmesan, 1/2 teaspoon of the salt and a generous grind of pepper. Blend to form a paste, scraping down the sides as needed. While the machine is running, drizzle in 1/2 cup oil. Add the hot water.
Sauté the fish fillets. Pat them dry with a paper towel and season with salt and pepper. Heat a tablespoon or so of olive oil in a nonstick skillet over medium heat. Add the fillets and cook for 3 minutes. Turn and cook until cooked 3 minutes. Transfer to a plate and let the fish rest for a few minutes and then break them into bite-size pieces with a fork.
Meanwhile, in a large pot of boiling, salted water, cook the linguine until just done, following package instructions. Reserve about 1/2 cup of the pasta water. Drain the pasta and transfer it to a medium bowl. Toss with the sauce, most of the fish chunks, 1/4 cup of the reserved pasta water and the remaining Parmesan. If the sauce seems too thick, add more of the reserved water (we didn’t need any extra).
Divide among shallow pasta bowls and top with the parsley and the remaining chunks of fish. Serve.