Preheat the oven to 400°. Grease a 9-by-13-inch baking dish with 1 tablespoon of the butter.
In a large pot of salted boiling water, cook the linguine until barely al dente, about 7 minutes. Drain well, transfer to a large bowl and toss with 1/2 tablespoon of the olive oil.
Meanwhile, arrange the chicken breasts, garlic cloves and bay leaves in a medium enameled cast-iron casserole; season with 1 tablespoon of salt and add enough water to cover the chicken by 1 inch. Cover the casserole and bring to a boil over moderately high heat, then reduce the heat to low and simmer for about 10 minutes, until the chicken is white throughout. Transfer the poached chicken to a plate and let cool slightly. Discard the garlic cloves, bay leaves and poaching liquid and wipe out the casserole. Tear the chicken into large chunks and add it and any accumulated juices to the bowl with the linguine.
In the casserole, heat the remaining 2 tablespoons of olive oil until shimmering. Add the onion and a generous pinch each of salt and pepper and cook over moderately high heat, stirring occasionally, until softened and lightly golden brown, about 6 minutes. Add the mushrooms and another pinch each of salt and pepper and cook until browned and softened, about 5 minutes. Transfer the vegetables to the bowl with the chicken and linguine. Wipe out the casserole.
In the casserole, melt 3 tablespoons of the butter over moderate heat. Add the flour, 1 teaspoon of salt and 1/4 teaspoon of pepper and cook, stirring constantly, until a smooth paste forms, about 1 minute. Add the milk in a slow, steady stream, whisking constantly. Bring to a boil, stirring frequently, then reduce the heat to low and simmer until the sauce is thick enough to coat the back of a spoon, about 12 minutes. Remove from the heat and whisk in the sour cream and mustard. Add the mozzarella, Fontina and Parmigiano-Reggiano and stir until melted. Scrape the cheese sauce into the bowl with the chicken, vegetables and linguine, then stir in the peas. Transfer the chicken mixture to the prepared baking dish.
In a small microwavable bowl, melt the remaining 2 tablespoons of butter. Stir in the panko and a pinch each of salt and pepper. Sprinkle the panko over the chicken and bake for 30 to 35 minutes, until the tetrazzini is bubbling and golden brown on top. Let cool slightly before serving.
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