Bring a large pot of salted water to a boil, add linguine and cook until a little less than al dente, about 8 minutes. Season chicken breasts with salt and pepper. Heat a deep skillet over medium-high heat. Add oil and 1 tablespoon of butter, heat until the butter is melted. Then add the chicken breasts, skin side down and cook for 2 to 3 minutes on each side, or until lightly colored.
When done, transfer chicken breasts to a plate and set aside. Discard any fat from your skillet. Now, to the skillet add 2 tablespoons of butter, heat until butter is melted. Add onion, celery, carrots and salt and pepper. Cook, stirring until vegetables are softened, about 3 minutes. Return chicken to the skillet, add rosemary, and bay leaf and cook, covered over moderately low heat for 5 minutes more or until chicken is cooked through but still slightly springy to the touch. Again, remove chicken from the skillet and transfer to a plate, pouring any remaining juice into the skillet and set aside. Now, prepare the sauce. To the same skillet add madeira and chicken broth, cook until reduced by half. Then add heavy cream and reduce until sauce is lightly thickened. Drain and pat dry linguine and transfer to a serving plate and toss with remaining tablespoon of butter. Top with chicken breasts. To the sauce add minced fresh herbs, taste for seasoning, adding salt, pepper and lemon juice to taste. Gently pour sauce over linguine and chicken, making sure to evenly coat the entire surface. Serve immediately.
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