Melt the butter in a large skillet over medium heat. Add the onion, carrots, and celery and cook for 5 minutes, stirring often, until beginning to soften.

Sprinkle the flour over the vegetables and stir for 1 minute. Gradually whisk in the chicken broth and milk. Add the salt, pepper, and dried thyme. Simmer for 4 to 5 minutes, whisking, until thickened.

Fold in the cooked chicken and frozen peas. Remove the skillet from the heat and cool the filling for 15 minutes so it does not soften the crust.

Heat the oven to 400°F. Spoon the filling into a 9-inch deep-dish pie plate. Unroll the pie crust over the top, trim the excess, and crimp the edges to seal.

Whisk the egg and water together. Brush the top crust with egg wash and cut 4 small vents in the center.

Bake for 30 to 35 minutes, until the crust is deeply golden and the filling bubbles through the vents. Rest for 10 minutes before serving.