1. At least 3 1/2 hours (or up to one day) before the filling is ready, roll puff pastry out on a lightly floured surface to a 3/16-inch-thick square. Place on a baking sheet, wrap tightly with plastic, and chill in the refrigerator for at least one hour.

2. Remove chilled dough from refrigerator, and place on a cutting board. Directly onto the dough, invert a bowl or place a lid that is 2 1/2 to 3 inches larger in diameter than a 3-quart round casserole or souffle dish. With a sharp knife, cut around the bowl or lid so that you have a perfect circle. Return the circle of dough to the baking sheet, wrap tightly with plastic, and chill for at least 2 hours.

3. Place a rack in lower two-thirds portion of the oven. A convection oven is ideal; preheat it to 375 degrees. If you don't have a convection oven, preheat a conventional oven to 425 degrees. In a large heavy-bottom pot, melt 5 tablespoons butter over medium-high heat. Add onion, potatoes, and carrots. Cook, stirring occasionally, until potatoes and carrots are fork-tender, 10 to 12 minutes. Add mushrooms, cooking until mushrooms are heated through. Add cognac, and cook for 30 seconds. Add flour, and cook, stirring, for 1 minute. Pour in chicken stock and milk. Bring to a simmer; cook until thick and bubbly, stirring constantly, 2 to 3 minutes. Stir in chicken, peas, parsley, and thyme. Season with salt and pepper. Transfer to 3-quart casserole or souffle dish. Transfer to a rack to cool slightly, about 5 minutes.

4. Make egg wash: In a small bowl, whisk together egg, egg yolk, and 1 teaspoon water. Remove dough circle from refrigerator, and brush surface evenly with egg wash. Invert dough and place directly over the casserole, pressing lightly to seal overhanging crust to the side of the dish. Brush top surface of dough with egg wash. With a 1/2-inch large round pastry tip (Ateco #7), cut a small circle in the center of the pastry, removing the pastry circle, and insert the pastry tip, narrow side down, into cut-out hole. This allows the steam to vent.

5. Transfer casserole to a parchment-lined baking sheet. Bake for 10 minutes; cover loosely with aluminum foil to prevent crust from burning, and continue baking for 25 minutes more. Serve hot.