For the casserole: Heat a few inches of water to a low boil, add salt and the green beans and cook 3 to 4 minutes to tender-crisp, drain.
Heat the EVOO in a large, deep skillet over medium to medium-high heat, add the mushrooms and lightly brown them, 7 to 8 minutes. Add the butter, shallots and garlic, season with salt, pepper and thyme, and stir 2 to 3 minutes more. Add the flour, stir a minute, add the wine, stir in the stock and thicken. Add the cream and Boursin and reduce the heat to simmer. Add the green beans and chicken and thicken a bit; stir in the tarragon.
Transfer into a casserole dish and cool. Cover and refrigerate for a make-ahead meal.
To reheat: Bring back to room temp. Bake at 375 degrees F until bubbly, 50 to 60 minutes; top with homemade or store-bought Fried Onions.
Place the chicken in a large stockpot. Add the celery, carrot, onion, garlic, lemon, bay leaf, peppercorns and the herb bundle; season with salt. Cover the chicken with water and bring to a boil; reduce the heat to a low, rolling simmer. Simmer 60 to 75 minutes, then cool the chicken in its stock. Strain the stock. Remove the chicken in large pieces from the skin and bones. Cut half of the meat into bite-size chunks. Thinly slice or pull the remaining meat.
Heat 2 to 3 inches of frying oil to 365 degrees F in a countertop fryer or in a deep medium pot over medium to medium-high heat. Separate the onion rings and soak in the buttermilk. Toss a few rings at a time into the flour and fry until crisp; drain on paper towels and repeat with the remaining onions.