For the chicken: Place the chicken in a pot and add the bay leaves, carrots, celery, lemons, onions, peppercorns, rosemary, thyme and some salt. Cover with water and bring to a boil. Reduce the heat to a simmer and poach the chicken for 1 hour. When the chicken is cool enough to handle, discard the skin and bones and separate the meat into bite-size pieces. Strain the stock and reserve.
For the pasta: Meanwhile, heat the butter and EVOO in a saucepan over medium-low to medium heat. Add the fennel, garlic and onions and cook very slowly until they are very sweet and soft, 20 to 25 minutes (do not let the onions brown). Deglaze the pan with the wine, scraping up any browned bits with a wooden spoon.
Transfer the fennel mixture to a blender or food processor and add the heavy cream and 1 cup reserved poaching liquid (or chicken stock). Puree until smooth, then return the mixture to the pot, season with salt and white pepper and simmer over low heat to thicken the sauce a little, 10 to 15 minutes. Add one-third of the chicken to the sauce and reserve the remaining chicken for another use.
Bring a large pot of water to a boil for the pasta and season with salt. Cook the pasta until al dente. Scoop out a bit of the cooking liquid and add it to the sauce to thin it slightly. Drain the pasta, then add it to the sauce and toss to combine. Garnish the pasta with the pine nuts, cheese, fennel fronds and tarragon.
If you are not serving the chicken immediately, cool the chicken-sauce mixture and refrigerate. Reheat in a saucepan over medium-low heat.