In a small bowl, combine ground coriander, turmeric, sumac, garlic powder, cumin, Kosher salt, sweet paprika, ground cardamom, ground cinnamon, and a pinch of Cayenne pepper. Mix well.
Drizzle the mushrooms with olive oil, then sprinkle the shawarma spice blend on top. Toss to thoroughly coat.
Heat a large cast-iron skillet over medium-high heat. When hot, add olive oil, let it heat up, then add the mushrooms. Weight them down with a food weight or another large heavy pan. Cook.
While the mushrooms are cooking, combine mayo, sour cream, white vinegar, garlic, sugar, Kosher salt, and black pepper. Whisk until smooth, then add a few teaspoons of water, as needed, to reach desired consistency for drizzling.
Pile onto warm pita bread with tzatziki (or white sauce), chopped lettuce, cucumbers, tomatoes, and pickled red onions. Drizzle with tahini, then serve with lemon wedges.