Poached chicken: Wash and dry the chicken and season the inside and outside with salt inside. Arrange the vegetables in bottom of a large pot and put the herbs and thick slices of lemon on top. Set chicken on top of the vegetables and cover with water. Bring the water to a boil over medium heat, then reduce the heat and gently poach chicken at a low simmer for 1 hour. Occasionally, skim off any foam that rises to the top. Turn off heat and let the chicken stand covered for 1 hour more. Remove the chicken from pot to a platter. Strain the liquid, and add about 3 1/2 quarts back to the pot. Reduce the liquid over a low flame for about 1 hour more, producing about 2 quarts of strong flavored stock. Remove the skin of the chicken. Pull the meat from the bones, then dice or shred the meat with forks. Refrigerate the stock and chicken until ready to use.
Yield about 2 1/2 to 3 pounds picked meat and 2 quarts concentrated stock
Drop Biscuit Chicken Pot Pie: Heat a high sided skillet over medium heat with extra-virgin olive oil, 2 turns of the pan, add the mushrooms and cook to tender, but not brown, stirring frequently, about 8 minutes. Add carrots, potato, celery, onion, salt and pepper, to taste. Cover the pan, and cook for 10 minutes more, stirring occasionally.
Meanwhile, heat a pot over medium-high heat and add the butter. Whisk in the flour, then whisk in the stock to form a roux. Season with salt and pepper and cook for 7 to 8 minutes to thicken and lightly coat the back of a spoon. Stir in the tarragon and Dijon mustard.
Pour the sauce into the vegetables and stir. Add the chicken and peas and transfer the mixture to a casserole dish or individual crocks. This meal can be refrigerated, at this point, for make-ahead dinners.
Preheat the oven to 375 degrees F.
Prepare the drop biscuit mix for 8 biscuits and stir in a little chives, parsley or dill for flavor. Top chicken casserole or crocks with drop biscuits and arrange them on a baking sheet. Bake in center of oven at 375 degrees F until casserole is warm through and biscuits are lightly golden and crispy at edges. If the chicken filling is still warm from stove top this will take only about 10 minutes. If the chicken is cold, bake for 30 minutes; for the last 10 minutes, remove from oven, add biscuit mix, then return to the oven and cook uncovered until the biscuits are golden.