Make the Spice Blend: Whisk all spice blend ingredients together in a small bowl until fully combined. Store any extra in a sealed spice jar for future use.
Caramelize the Onions: Heat olive oil in a medium heavy-bottomed pan over medium heat. Add sliced onion with a pinch of salt and cook until translucent. Stir in sweetener, cover with a lid, and cook on medium-low heat, stirring often, until deeply golden and jammy - about 15–20 minutes. Transfer to a bowl and set aside.
Cook the Mushrooms: In the same pan, increase heat to medium-high. Add shredded oyster mushrooms with a splash of white wine or lemon juice. Stir in fresh thyme leaves and cook until mushrooms are wilted and liquid has mostly evaporated.
Season: Add 3 teaspoons of the gyros spice blend to the mushrooms. Continue cooking, stirring frequently, until all moisture has evaporated and the mushrooms look slightly crispy. Taste and adjust seasoning with sea salt as needed.
Optional Broil Step (Highly Recommended): Transfer mushroom filling to a broiler-safe dish. Lightly spray with olive oil. Broil for 3–4 minutes until edges are crispy. Remove from heat and stir in the reserved caramelized onions.
Warm the Pita: Heat pita breads in the microwave for 30 seconds, under the broiler for 1–2 minutes, or in a dry skillet until warm and pliable.
Assemble: Fold each pita into a cone shape and wrap the bottom with parchment paper, securing with kitchen twine if desired. Fill generously with the mushroom and onion mixture. Top with cucumber slices, tomato, red onion, and fresh dill. Drizzle generously with vegan tzatziki sauce.
Serve immediately and enjoy!