Preheat oven to 400°F.
In a Dutch oven, heat 1 tablespoon of olive oil over medium heat. Add mushrooms and season with salt and pepper. Sauté for 5 minutes or until they start to brown. Remove mushrooms from pan and set aside.
Pour 1 tablespoon olive oil into the pot and add 3 chopped shallots and 3 sliced carrots. Season with salt and pepper. Cook for 7 minutes stirring occasionally.
Add to the pot, 3 minced garlic cloves, 1 tablespoon of minced thyme, 1 teaspoon paprika, 2 tablespoons tomato paste, and 2 bay leaves. Cook for 1 minute.
Add mushrooms back to the pot and vegetable stock. Bring to a boil for 2 minutes. Reduce the heat and simmer for 25 minutes. Once cooked, stir in 2 tablespoons arrowroot powder, and remove the bay leaves.
Roll out puff pastry to the size of a pie dish. In a greased pie dish add mushroom filling. Top the dish with the puff pastry. Brush the pastry with non-dairy milk. Cut three vents to the center of the puff pastry.
Cook for 25 minutes until pastry is golden brown. Sprinkle with fresh parsley.