Place the chicken pieces, diced carrots, diced celery, diced onion, minced garlic, and frozen peas into the slow cooker.
In a medium saucepan, melt the butter over medium heat. Once melted, whisk in the flour and cook for 1 to 2 minutes until it forms a smooth paste and is lightly golden.
Gradually whisk in the chicken broth and milk, stirring constantly to avoid lumps. Continue cooking and stirring until the sauce thickens, about 3 to 5 minutes.
Add the dried thyme, salt, and black pepper to the sauce. Pour the sauce over the ingredients in the slow cooker and stir gently to combine.
Cover and cook on low for 4 to 4.5 hours, until the chicken is cooked through and the vegetables are tender.
About 30 minutes before serving, preheat the oven to 400°F (200°C).
Transfer the chicken and vegetable mixture from the slow cooker into a 9-inch pie dish or oven-safe casserole dish.
Unfold the thawed puff pastry sheet and place it over the top of the filling, trimming any excess pastry hanging over the edges. Press the edges to seal.
Brush the beaten egg over the puff pastry to give it a golden color when baked.
Bake in the preheated oven for 20 to 25 minutes, or until the puff pastry is puffed and golden brown.
Remove from the oven and let the chicken pie rest for 5 minutes before serving.