Put all tzatziki ingredients in a bowl, combine them well and place in the fridge;
To make the Greek pitas, put the yeast in a bowl and combine with 70 ml of warm soy milk; set aside for 10 minutes;
Meanwhile put the flour, oil and salt in a large bowl and combine well; add the yeast mixture and start kneading; gradually add the remained milk and knead until you get an elastic dough; knead for another 3-4 minutes then transfer it to a bowl greased with oil; cover the bowl with a clean towel and let it rise for about an hour or until it doubles in size;
Put the mushrooms in a bowl with 1 tbsp of oil and spices; combine well and set aside for 30 minutes;
Heat a large pan or a wok and add 120 ml water and the mushrooms, cover with a lid and sauté for 30 minutes; stir from time to time; at the end, add 1-2 tbsp of oil and cook over high heat for 2 minutes;
When the dough doubles in size, remove it from the bowl and place it on a surface dusted with flour; cut the dough into 4 equal parts, shape them into balls and allow them to rest for 10 minutes;
Use a rolling pin to roll each dough ball into 23-24 cm discs;
Heat a pan, grease it with a bit of oil if needed, and cook the pitas for about 2-3 minutes on each side over medium heat;
To assemble the gyros, put on each pita lettuce leaves, then a mushroom layer, fresh veggies and cashew tzatziki sauce on top.