Preheat the oven to 425 degrees F (220 degrees C).

Melt butter in a large, deep skillet over medium-high heat. Add onion, carrot, celery, salt, and rosemary and cook, stirring often, until onion is translucent, about 5 minutes. Stir in flour and cook for 1 minute.

Stir in chicken broth and cream and bring to a simmer. Reduce heat and cook, stirring often, until thick and creamy and vegetables have started to soften, about 5 minutes. Stir in chicken, peas, and parsley until well combined. Season with salt and pepper and remove from the heat.

Place one crust in the bottom of a 9-inch pie plate. Add filling and place second crust on top. Press crusts together to seal around the edges and use fingers to crimp the dough. Cut small slits in the top and transfer to a rimmed baking sheet.

Bake in the preheated oven until golden and bubbly, about 30 minutes. Tent the edges with foil while baking if they become too brown. Let stand 10 minutes before serving.