Prep: Preheat the oven to 400F and set a baking sheet and a 10” oven-safe cast-iron pan (or similar-sized baking dish) aside. Thaw the puff pastry according to package instructions.
Aromatics: Warm the oil in the oven-safe sauté pan over medium-high heat. Add the onion, mushrooms, and carrots. Sauté for 7 minutes, stirring occasionally, until the vegetables are lightly browned. Stir in the tomato paste and sauté for an additional 2 minutes. Add the garlic, thyme, paprika, and black pepper; sauté for 30 seconds, or until fragrant
Deglaze: Deglaze the pan with the red wine (or sub water) and cook until evaporated, 2 to 3 minutes. Sprinkle the flour over the vegetables and sauté for 1 minute.
Simmer: Add the soaked cashews, vegetable broth, Worcestershire, and tamari to a blender; blend for 45 to 60 seconds, until smooth. Add this mixture and the potatoes to the pan and stir well. Bring the filling to a boil over high heat, then reduce the heat to medium and simmer for 10 to 12 minutes, stirring occasionally. Stir the frozen peas into the mixture and remove from the heat.
Assemble: *If you are using a different baking dish, transfer the casserole filling now. If necessary, roll your puff pastry out so it can cover the pan. Transfer the puff pastry over the top of the pan and trim the edges so there is a slight 1” overhang. Fold the extra pastry under into the pan and carefully pinch the pastry to secure it to the edges of the pan. Use a knife to create 6 1-inch slits in the pastry for ventilation (see blog post for photos). Brush the top of the dough with the remaining cashew milk.
Bake: Place the pot pie on top of the baking dish to catch spills, then place in the middle rack of the oven. Bake for 25 to 30 minutes, until the crust is golden and the filling is bubbly. Let the mushroom pot pie cool (and set) for 10 minutes before serving. Serve warm; leftovers can be stored in the refrigerator for up to 5 days, and reheated in the oven, toaster oven, or microwave.