Place the chicken pieces, diced onion, sliced carrots, sliced celery, 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 2 minutes, stirring constantly to form a roux.
Gradually whisk in the chicken broth and milk, continuing to whisk until the mixture thickens and becomes smooth, about 3-5 minutes.
Add the dried thyme, dried rosemary, salt, and black pepper to the sauce. Stir to combine.
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.
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 similar oven-safe dish.
Roll out the thawed puff pastry sheet to fit over the top of the pie dish. Place the pastry over the filling and trim any excess pastry from the edges.
Brush the beaten egg over the puff pastry to give it a golden finish.
Cut a few small slits in the pastry to allow steam to escape.
Bake in the preheated oven for 20-25 minutes, or until the puff pastry is golden brown and puffed.
Remove from the oven and let cool for 5 minutes before serving.