Melt the butter in a large skillet or Dutch oven over medium-high heat. Add the onion, carrots, and celery and cook for 6 minutes, stirring occasionally, until softened. Add the garlic and stir for 30 seconds until fragrant.
Sprinkle the flour over the vegetables and stir to coat. Cook for 1 full minute to toast out the raw flour taste. Pour in the chicken broth gradually, whisking as you go to prevent lumps. Add the heavy cream, salt, pepper, thyme, and parsley.
Simmer over medium-low heat for 8–10 minutes, stirring occasionally, until the sauce is thick enough to coat the back of a spoon. Remove from heat and fold in the shredded chicken and frozen peas. Taste and adjust salt.
Preheat oven to 400°F if baking one tonight. Press one round of pie crust into a 9-inch pie pan. Spoon half the filling in. Top with a second crust round, crimp the edges, and cut 4–5 steam vents. Repeat for the second pie.
For the pie baking now: brush the top crust with egg wash. Bake on a rimmed baking sheet at 400°F for 35–40 minutes until deep golden and the filling bubbles through the vents. Rest 10 minutes before slicing.
For the pie going in the freezer: do NOT brush with egg wash. Cover tightly with foil then plastic wrap. Freeze for up to 3 months. To bake from frozen: bake covered at 400°F for 30 minutes, remove foil, brush with egg wash, and bake 30–40 more minutes until golden and bubbling. Internal temperature should reach 165°F.