Place potatoes in a large pot of well-salted water and bring to a boil over high heat. Reduce heat and simmer gently until tender, about 20 minutes.
While potatoes simmer, trim, bone, and cut the cod fillets across into 2-inch pieces.
Drain potatoes well and return to the pot. Add a pinch of cayenne, salt, and 4 tablespoons butter. Mash until smooth.
Preheat the oven to 450 degrees F (230 degrees C). Grease 2 individual-sized casserole or baking dishes with 1 tablespoon butter. Place both dishes on a baking sheet.
Transfer about 2/3 cup of the mashed potatoes into each of the prepared dishes. Smooth out evenly and make a shallow depression with a spoon down the middle. Set aside.
Heat remaining 2 tablespoons butter in a deep 10-inch skillet over medium-high heat. Cook the shallots in the hot butter with a pinch of salt until they just start to turn golden, 3 to 4 minutes. Add cream, lemon zest, and a pinch of cayenne, and bring to a simmer. Add cod and wait for the sauce to come back to a simmer. Cook for 1 minute, while basting with the sauce; turn and cook for 1 minute more.
Pour in lemon juice and turn off the heat; stir for about 30 seconds.
Divide the pieces of cod over the potato. Add enough sauce to just cover the fish and return the pan to the stove.
Bring sauce to a boil over medium-high heat and cook until the sauce reduces and thickens, 1 to 2 minutes. Turn off the heat and stir in tarragon.
Spoon the thickened sauce over the cod. Add a light dusting of Parmigiano-Reggiano cheese.
Bake in the center of the preheated oven until cod is cooked through and the sauce is browned and bubbling, 10 to 15 minutes. Let rest for 5 minutes before serving.
Use cod fillets that are 6 to 8 ounces each.
You can add as much potato as you want to the casserole dishes but remember to leave room for the fish and sauce.
This recipe will make extra mashed potatoes; 1 large russet potato will be enough for 2 portions. Recipe can be tripled and baked in a 9×13-inch casserole dish.
Substitute dill for tarragon if desired.
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