In a large pot over medium heat, heat the oil. Add the onion and carrots and cook 5 minutes, or until the onions begin to turn translucent.
Add the garlic, celery, and mushrooms and cook another 5 minutes, or until tender.
Add the salt, pepper, Italian seasoning, and arrowroot starch. Stir to combine.
Add the lemon juice, vegetable broth, almond milk, broccoli, and chicken. Stir to combine. Add more arrowroot, if necessary, to reach desired thickness. Remove mixture from heat and allow to cool.
Preheat the oven to pie crust or biscuit package specification.
To make one family-style pie: Place one pie crust into an ungreased 9-inch (23-cm) pie dish. Fill with the vegetable mixture. Top with the second pie crust, folding under any excess crust and crimping the edges with a fork.
Brush the crust evenly with beaten egg. Cut a few slits in the top crust.
Bake according to pie crust packaging.
Cut and serve warm.
To make individual pies: Arrange 6 or more ramekins or bowls on a baking sheet.