Preheat the oven to 400°F. Grease a 3-quart casserole with butter. Cut the poached chicken into ¾-inch pieces.

Melt the butter in a large stockpot over medium heat; add the onion and sauté, stirring occasionally, until tender, 8 to 10 minutes. Whisk in the flour and, stirring constantly, cook for about 2 minutes. While whisking, slowly pour in the chicken stock and then the milk. Turn up the heat to medium-high and continue whisking for a few minutes until the sauce is thick and  creamy.. Turn off the heat.

In a saucepan of boiling water, cook the carrots until fork-tender but not overcooked, 6 to 8 minutes; drain and stir into the sauce along with the chicken, mushrooms, peas, and tarragon. Adjust the seasonings to taste with salt and pepper. . Stir in the sherry. The sauce should be thick and creamy, not pasty; if necessary, stir in a little more milk. Spoon the mixture into the buttered casserole.

Cut the pie crusts into 2 large rectangles that, side by side, completely cover the casserole and drape about 1 inch over the sides; if necessary, roll out the crusts a little on a clean floured surface. Press the dough into place. In a small bowl, beat together the egg and water and brush over the pastry. With a fork, punch a dozen or so holes all over the crust. Bake uncovered in the preheated oven until golden brown, 25 to 30 minutes. Remove from the oven, rest at room temperature for 5 minutes, and use a large spoon to serve individual portions of crust and filling.