Preheat the oven to 350°F.
Melt the butter in a large, heavy-bottomed pot over medium heat, then add the onion and garlic. Sauté for 4 to 5 minutes, until the onion is translucent and fragrant.
Sprinkle the flour over the onion mixture and whisk to combine.
Add the chicken broth 1 cup at a time, whisking constantly, then stir in the heavy cream, sage, thyme, salt, and pepper, continuing to whisk until mixture is smooth.
Stir in the chicken and the peas and carrots. Simmer the mixture for 5 minutes.
Transfer the chicken mixture to a deep pie pan or 2-quart casserole dish, then lay the crust over the top. Brush with the whisked egg, then bake for 30 minutes.
Remove from the oven and let cool slightly, then serve!