FOR THE PASTRY: Place the flour in a bowl. Using a pastry blender or your fingertips, work in the butter until crumbly. Add the cheese and work in until just blended. Sprinkle the ice water over the pastry dough, a little at a time, and gather the pastry into a ball. Knead lightly until just combined. Wrap in plastic wrap and chill until needed. FOR THE FILLING: In a saucepan over medium heat, bring the stock to a simmer. Add the chicken and simmer, uncovered, until opaque throughout, 15-20 minutes. Remove from the heat and let the chicken cool completely in the liquid. Remove the breasts, reserving the stock. You should have about 2 1/2 cups stock. Cut the chicken into 3/4-inch chunks. Set aside. Bring a saucepan three-fourths full of water to a boil and salt lightly. Add the carrots and cook over medium-high heat for 5-6 minutes. Add the celery, pearl onions and peas and cook until all are barely tender, about 3 minutes longer. Drain well; set aside. In a saucepan over medium heat, melt the butter. Sprinkle in the flour and whisk until the mixture is gently bubbling and smooth, 2-3 minutes; do not brown. Gradually add the reserved stock, whisking constantly, and bring to a simmer. Cook, stirring often, until smooth and slightly thickened, 4-5 minutes. Add the cream and cook, stirring occasionally, until the sauce coats the back of the spoon, about 5 minutes longer. Remove from the heat and stir in the salt and pepper to taste, the thyme, chives and parsley. TO ASSEMBLE: Preheat an oven to 400 degrees F. Add the chicken and vegetables to the sauce and stir to combine. Spoon into a 9-by-13-inch baking dish. Brush the edge of the dish with some of the beaten egg. On a lightly floured work surface, roll out the pastry into a rectangle 10 by 15 inches. Transfer the pastry to the dish, pressing on the edges firmly. Trim away the overhang. Gently knead the dough scraps together, roll out 1/8 inch thick and cut out several small leaf shapes. Brush the top of the pie with the beaten egg. Using the knife, score the pastry leaves lightly, attach them to the pie pastry, and brush with more egg. Cut 3 slits each 1 inch long at the center of the pie. Bake until golden, 25-30 minutes. Remove from the oven and let stand for 5 minutes, then spoon onto warmed individual plates or bowls.