1. Season chicken with 1/4 teaspoon each salt and pepper. Heat 2 table-spoons of butter in a large frying pan over medium heat. Add chicken and cook, turning, until browned outside with no trace of pink inside, about 10 minutes. With tongs, remove chicken from pan.
2. Add remaining 2 tablespoons of butter to pan, reduce heat to medium-low, and add leeks, mushrooms, and tarragon. Cook, stirring occasionally, until mushrooms are tender and leeks are soft, about 7 minutes. Return chicken to pan and add cream and Madeira. Bring to a boil, reduce heat, and simmer, uncovered, 3 minutes to reduce sauce slightly. Season with remaining 1/4 teaspoon each salt and pepper.
3. Meanwhile, in a large pot of boiling salted water, cook pasta until tender but still firm, 5 to 7 minutes. Drain in a colander.
4. Toss pasta with chicken and mushroom sauce. Divide among 4 dinner plates and sprinkle with parsley.