For the pastry, place flour in a mixing bowl and season lightly with salt, about 1/4 teaspoon. Add sage, butter and milk or half-andhalf, and combine to form dough.
For the filling, poach chicken in a pot with carrot, celery, onion, bay, herb bundle and salt. Add enough water to just cover the chicken, bring to a gentle boil, reduce heat and simmer about 40 minutes. Remove breast and cool to handle. Strain stock and keep at a low simmer. Remove chicken from bones, discard skin and bones, and pull or chop chicken into bite-sized pieces.
Heat EVOO, 1 turn of the pan, in a large, deep skillet or Dutch oven over medium-high heat. Add butter and melt into oil then add mushrooms and brown until golden, 15 minutes. Season mushrooms with salt and pepper, add shallots and stir 2 minutes more. Add beans and toss together, sprinkle flour over pan, stir a minute more then add wine and reduce 1 minute. Ladle in about 3 cups of warm chicken stock and let thicken a bit. Stir in about 1/4 cup heavy cream, add tarragon and stir in Dijon. Season with a few grates of nutmeg, add chicken and combine.
Preheat oven to 400F.
Transfer chicken to casserole dish. Roll out dough and trim to fit the casserole. Cut a few slits in dough to allow steam to escape and brush with egg wash. Bake until deeply golden, 30-35 minutes.