Melt the butter in a large skillet over medium heat. Add the onion, carrots and celery and cook, stirring now and then, for 7 to 8 minutes, until the onion is soft and translucent. Add the garlic and cook 30 seconds.
Sprinkle in the flour and stir constantly for 1 to 2 minutes. Slowly pour in the broth while whisking, then the milk. Add the thyme, sage, salt and pepper.
Simmer, stirring often, for 3 to 5 minutes, until the sauce is thick enough to coat the back of a spoon. Taste and adjust the salt. Stir in the chicken and frozen peas.
Take the pan off the heat and let the filling cool for 20 to 30 minutes, until just warm.
Heat the oven to 400°F (200°C) with a rack in the lower third. Fit one pie crust into a 9-inch deep-dish pie plate.
Pour in the cooled filling. Lay the second crust on top, trim the overhang to about 1 inch, fold it under and crimp the edges. Cut 4 small slits in the top.
Beat the egg with the milk and brush it over the top crust. Set the pie on a baking sheet.
Bake for 35 to 40 minutes, until the crust is deep golden and the filling bubbles through the slits. Cover the edges with foil after 20 minutes if they brown too fast.
Let the pie rest for 15 minutes before cutting.