How to Make It
In a large frying pan, heat 2 tablespoons of the oil over moderately high heat. Add the fennel, 1 teaspoon salt and 1/2 teaspoon pepper. Cook, stirring frequently, until the fennel is soft and golden, 8 to 10 minutes.
In a medium frying pan, heat the remaining tablespoon oil over moderately high heat. Season the chicken with 1/8 teaspoon salt and a pinch of pepper and add it to the pan. Cook the chicken until brown, about 3 minutes. Turn the chicken, cover the pan and cook over moderate heat until just done, about 5 minutes longer. Remove the chicken, let it rest 5 minutes and then tear the meat into large shreds.
Add the stock and wine to the pan, along with the dried tarragon, if using. Boil, scraping the bottom of the pan to dislodge any brown bits, until reduced to about 1/2 cup. Remove the pan from the heat and whisk in the butter and fresh tarragon, if using.
In a large pot of boiling, salted water, cook the linguine until just done, about 12 minutes. When the pasta is almost done, add the fennel and chicken to the sauce, with any juices that may have accumulated on the plate, and reheat.
Drain the linguine and toss it with the chicken mixture. Top with a few fennel sprigs.