Make a roux by whisking 5 tablespoons butter and 5 tablespoons flour in a small bowl until smooth.
Melt remaining 2 tablespoons butter in a large deep skillet or wide pan over medium-high heat. Add onion, celery, and carrots and sauté until vegetables are softened and start to brown, about 10 to 12 minutes. Add thyme, garlic and dried sage and cook for 2 minutes, stirring.
Add 1/3 cup dry white wine and boil until almost fully evaporated, about 2 to 3 minutes. Add 2 cups chicken broth and bring to a simmer. Whisk in the butter-flour roux a spoonful at a time until incorporated and mixture boils. Reduce heat to medium and simmer gently until sauce thickens, stirring frequently, about 3 to 5 minutes.
Stir in the cubed chicken, peas, cream, and parsley and bring to a simmer. Season with salt and pepper. Remove from heat and let filling cool.
Preheat oven to 425°F. Transfer filling to bottom pie shell. Cover with top crust, seal the edges and trim away any excess dough. Slash top crust with 2 to 4 small slits for steam to escape. Brush top and edges of the crust with the beaten egg. Bake 35 to 40 minutes until pie is heated through and crust is deep golden brown. You may need to cover the pie edges in foil to prevent over-browning in the last 10 or 15 minutes. Let rest and cool for 10 minutes before serving.