Set the chicken in a deep soup kettle or large casserole breast side down and add the stock or broth and water and 4 teaspoons of salt. Bring to a boil over high heat, skimming the surface thoroughly. Lower the heat and add the carrots, celery, leeks, bouquet garni, thyme, garlic, and dried tarragon. Adjust the heat to a low simmer and poach the chicken for 15 minutes. Turn the chicken onto its back and continue poaching for 25 minutes more, or until it is cooked through.
Transfer the chicken to a large plate. Strain the stock into a large saucepan. Reserve the vegetables and discard the bouquet garni. Set the stock over moderately high heat and reduce it by half.
Meanwhile, when the chicken is cool enough to handle, discard the skin and bones and shred the meat into large pieces. Set the chicken aside, covered, in a warm place.
In a saucepan of boiling water, add the pasta and cook for 5 minutes, or until al dente. Drain and set aside.
Whisk the butter into the broth and add the lemon juice and fresh tarragon. Taste and adjust the seasoning, adding more lemon juice, salt, and pepper. Add the vegetables to the saucepan along with the chopped parsley and pasta. Warm over low heat.
Make the garnish: Brush the bread with the oil and toast until golden. Sprinkle each slice with 2 tablespoons Parmesan and set a slice on each of four deep dinner plates.
Divide the chicken among the plates, spoon the broth, vegetables, and pasta over it and garnish each plate with the parsley sprigs.