To make the stew: Heat the butter in a 9-inch cast-iron skillet over medium-high heat. Add the carrot, celery, mushrooms, and salt and cook, stirring occasionally, until lightly browned, about 4 minutes. Add the onions and cook, stirring, until browned, about 1 minute more. Stir in the flour and cook, stirring, for 1 minute. Pour in the chicken broth, bring the stew to a boil, stirring constantly. Simmer until the mixture is very thick, about 3 minutes. Add the chicken and return to a boil. Remove from the heat. Season with pepper to taste and stir in the parsley, chives, and dill. Set aside while you make the biscuit topping.
To make the biscuit dough: In a large bowl whisk the flour, baking powder, sugar, and salt until evenly combined. Rub 2 tablespoons of the butter into the flour with your fingertips, until it is completely absorbed. Work the remaining butter into the flour until it is in even pieces about the size of a pea. Add the lemon zest to the milk. Gently stir the milk into the flour mixture to make a loose dough.
Lightly dust a clean work surface with flour and turn the dough out onto it. Pat the dough by hand into a 1/2-inch thick rectangle. Fold the dough in thirds like a business letter. After folding the biscuit dough into thirds, pat into a 10-inch wide disc.
Preheat the oven to 450 degrees F.
Bring the chicken and vegetable mixture to a boil and place the biscuit dough on top. Using a paring knife, make a small hole in the center of the dough. Using a pastry brush, brush the biscuit top lightly with milk. Transfer the skillet to a baking sheet and bake until golden brown, about 20 minutes. Let the pie rest at room temperature for 5 minutes before serving.
Put the parsley, thyme, onion, carrot, celery, and chicken breasts in a medium saucepan. Cover with the broth, and bring just to a boil. Lower the heat to very low and cover. Poach the chicken for 20 minutes or until firm to the touch. Remove the pan from the heat, uncover, and cool the chicken in the liquid for 30 minutes.
Transfer the chicken to a cutting board and reserve the liquid. Bone and skin the chicken and cut the meat into 3/4 inch cubes. Discard the bones and skin.
Strain the broth and store, covered, in the refrigerator for 3 days or freeze for later use. Remove any fat from the surface of the broth before using.
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