Preheat the oven to 400°F and set a baking sheet and a 10-inch oven-safe cast-iron pan or similar-sized baking dish aside. Thaw the puff pastry according to the package instructions.

Warm the olive oil in the oven-safe sauté pan over medium-high heat. Add the diced onion, quartered cremini mushrooms, chopped oyster mushrooms, and diced carrots. Sauté for 7 minutes, stirring occasionally, until the vegetables are lightly browned. Stir in the tomato paste and continue to sauté for an additional 2 minutes. Add the minced garlic, fresh thyme leaves, paprika, and black pepper and sauté for 30 seconds until fragrant.

Pour in the dry red wine to deglaze the pan, scraping any browned bits from the bottom. Cook until the liquid evaporates, about 2 to 3 minutes. Sprinkle the all-purpose flour over the vegetables and sauté for 1 minute to cook out the raw flour taste.

In a blender, combine the soaked cashews, vegetable broth, vegan Worcestershire sauce (if using), and tamari or soy sauce. Blend for 45 to 60 seconds until smooth and creamy. Add this cashew sauce mixture and the diced potatoes to the sauté pan. Stir well and bring the filling to a boil over high heat. Reduce the heat to medium and let it simmer for 10 to 12 minutes, stirring occasionally, until the potatoes are tender and the filling has thickened. Stir in the frozen peas and remove the pan from heat.

If using a different baking dish, transfer the filling now. Roll out the puff pastry if necessary to cover the pan with about a 1-inch overhang. Place the puff pastry over the filling and trim edges, folding the extra pastry under to secure to the pan's edges. Use a knife to cut six 1-inch slits in the pastry for ventilation. Brush the top of the pastry with the non-dairy milk for a golden finish.

Place the pot pie on top of a baking sheet to catch any spills, then place on the middle rack of the preheated oven. Bake for 25 to 30 minutes until the crust is golden brown and the filling is bubbly. Let the pot pie cool and set for 10 minutes before serving. Serve warm and enjoy!