Preheat the oven to 400F.
Preheat a 12” cast-iron or stainless-steel skillet over medium-high heat. Add olive oil and let heat for 60 seconds. Add carrot, celery, onion and the broccoli stems. Season with a pinch of salt and cook, stirring, until the onions are soft and translucent. Add garlic and cook 60 seconds.
Add arrowroot starch and cook, stirring, 60 seconds. Add the chicken stock and stir to incorporate. Add the coconut milk and bring to a gentle simmer.
Add the thyme and broccoli florets and stir to combine. Stir in the shredded chicken and bring everything to a simmer.
Remove the dough from the fridge and carefully lay it overtop of the skillet. Use a paring knife to cut any pie crust that drapes over the edge. Make an “X” in the center of the crust with the knife to allow some steam to release during cooking.
In a small bowl, whisk together one egg and one tablespoon of water. Brush the top of the pie crust with the egg wash mixture.
Place the skillet on a sheet pan to catch any drips and bake for 35 to 40 minutes or until the pie crust is golden brown.
Remove from oven and serve.