1 (4-pound) whole chicken (broth)
8 large sprigs fresh parsley (broth)
4 stalks celery (broth)
4 carrots, halved (broth)
1 medium yellow onion, cut into quarters (broth)
1 tablespoon black peppercorns (broth)
3 tablespoons unsalted butter (filling)
1½ cups (1-inch) sliced fresh green beans (filling)
1½ cups coarsely chopped yellow onion (filling)
1½ cups coarsely chopped carrot (filling)
1 cup coarsely chopped celery (filling)
2 cloves garlic, minced (filling)
¼ cup all-purpose flour (filling)
½ cup heavy whipping cream (filling)
2 tablespoons chopped fresh parsley (filling)
1 tablespoon chopped fresh thyme (filling)
2 teaspoons finely chopped fresh sage (filling)
2 teaspoons kosher salt (filling)
½ teaspoon ground black pepper (filling)
½ (17.3-ounce) package frozen puff pastry, thawed (crust)
1 large egg (crust)
1 teaspoon water (crust)
¼ teaspoon kosher salt (crust)
⅛ teaspoon ground black pepper (crust)
1. For broth: In a large stockpot, combine chicken, parsley, celery, carrot, onion, and peppercorns. Add water to cover by 1 inch. Bring to a simmer over medium-high heat. Reduce heat to medium, and simmer until chicken is tender, about 2 hours.
2. Remove chicken from broth. Let stand until cool enough to handle. Shred meat, discarding skin and bones. Drain broth through a fine-mesh sieve, discarding solids. Reserve 2 cups broth for filling. Let cool, and freeze remaining broth in heavy-duty resealable plastic bags for up to 3 months.
3. Preheat oven to 375°.
4. For filling: In a 10-inch cast-iron skillet, melt butter over medium-high heat. Add beans, onion, carrot, and celery; cook until softened, about 5 minutes. Add garlic; cook for 1 minute. Sprinkle flour over vegetables; cook for 1 minute, stirring constantly. Gradually stir in 2 cups reserved broth and cream. Cook, stirring constantly, until mixture begins to thicken. Remove from heat. Stir in chicken, parsley, thyme, sage, salt, and pepper.
5. For crust: On a lightly floured surface, roll puff pastry into a 10-inch square. Place over hot chicken mixture, letting excess extend over sides of skillet. In a small bowl, whisk together egg and 1 teaspoon water. Brush pastry with egg mixture. Cut 4 (1-inch) vents in top of dough to release steam. Sprinkle with salt and pepper. Brush underside of pastry corners with egg mixture; press to skillet.
6. Bake until filling is bubbly and crust is puffed and golden brown, about 25 minutes. Let cool for 15 minutes before serving.