Preheat the oven to 400°F. Lightly grease two 10-ounce ramekins.
Melt the butter in a skillet over medium heat. Add the onion, celery, and carrots. Cook for 5–6 minutes until the vegetables begin to soften. Stir in the garlic and cook for 30 seconds.
Sprinkle the flour over the vegetables and cook for 1 minute, stirring constantly.
Slowly whisk in the chicken broth, followed by the milk. Cook until the sauce thickens, about 3–4 minutes.
Stir in the chicken, peas, thyme, parsley, salt, and pepper. Simmer for 2 minutes, then remove from the heat.
Divide the filling evenly between the prepared ramekins.
Roll out the pie crust and cut two circles slightly larger than the tops of the ramekins. Place a crust over each ramekin and gently press around the edges. Cut a few small slits in the tops to vent steam.
Brush with the beaten egg.
Place the ramekins on a baking sheet and bake for 25–30 minutes, or until the crust is deep golden brown and the filling is bubbling.
Let stand for 5–10 minutes before serving.