Prepare the Dough: In a medium bowl, combine the warm water and soy milk (it should feel very warm but not too hot). Whisk in the sugar, then the yeast, and let it sit for about 5 minutes until frothy. Next, whisk in the olive oil and salt.
Add the flour and stir with a wooden spoon until a shaggy dough forms. Transfer the dough to a floured surface and knead for about 10 minutes until it becomes smooth and elastic. Place the dough back in the bowl, cover with plastic wrap, and let it rise in a warm spot until doubled in size, about 1 ½ hours.
Make the Bean Filling: While the dough is rising, heat a little oil in a skillet over medium heat. Add the chopped onion and cook for 7-8 minutes until soft and golden. Transfer to a bowl to cool.
In a food processor, pulse the walnuts until roughly chopped. Add the beans, cooked onions, garlic powder, smoked paprika, chili powder, cumin, mushroom powder, salt, and black pepper. Pulse until combined but still slightly chunky. Transfer to a bowl and set aside.
Shape and Fill the Dough: Once the dough has risen, turn it out onto a floured surface and form it into a circle. Divide the dough into 6 equal pieces. If the dough is sticky, dust your hands with flour or wet them with a bit of water.
Shape each piece into a small circle, gently stretch them, and fill with 3-4 tablespoons of the bean mixture. Pinch and seal the dough tightly around the filling, placing each filled pita onto a parchment-lined baking sheet. Cover with a tea towel and let rest for another hour.
Cook the Pitas: After the second rise, roll each filled dough ball into a 12 cm (5-inch) circle. Heat a cast iron or heavy-bottomed non-stick skillet over high heat until hot, then reduce to medium heat.
Place one dough round in the skillet, cooking for about 30 seconds on each side. Drizzle with a little olive oil (about 1 tbsp) and cook for another 5 minutes, flipping every minute until deep golden brown and puffed up. Repeat for all pitas.
Serve: Serve warm, topped with fresh oregano and a dollop of vegan tzatziki. Enjoy!