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1
Preheat oven to 400 degrees F (200 degrees C).
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2
Pour milk into a large skillet, place over medium heat, and place fish fillets into milk. Bring to a simmer and cook fish in milk for 1 minute. Transfer fillets to a plate and pour milk into a large measuring cup.
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3
Fill a large pot with lightly salted water and bring to a rolling boil. Stir in the spinach fettuccine, bring back to a boil, and cook pasta over medium heat until cooked through but still firm to the bite, about 8 minutes. Drain and transfer into a 2-quart casserole. Toss fettuccine with 1 tablespoon butter.
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4
Melt remaining 3 tablespoons butter in skillet over medium heat and whisk flour into butter, stirring constantly until mixture forms a paste. Cook and stir for 1 minute, then whisk reserved milk and chicken stock into flour mixture until thoroughly combined. Bring to a simmer, reduce heat to low, and simmer until sauce has thickened, about 5 minutes, stirring often. Remove sauce from heat; whisk in Dijon mustard, black pepper, nutmeg, 1/4 cup of Parmesan cheese, and 1/4 cup of Gruyere cheese.
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5
Spread spinach over fettuccine noodles; arrange fillets on top of spinach. Pour sauce over the fish and sprinkle remaining Parmesan and Gruyere cheeses over the top.
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6
Bake in the preheated oven until sauce is bubbling, about 15 minutes. Heat the oven's broiler and broil until cheese topping is browned, 1 to 2 more minutes.
