Place the soy curls in a large bowl and add in the broth, minced garlic, chili powder, oregano, salt, cumin, Stir to combine everything and press the curls down into the liquid. Allow the soy curls to rehydrate for 10-15 minutes.
Meanwhile, make the vegan tzatziki. Destem and finely chop the dill. In a small bowl, add the yogurt, chopped dill, minced garlic, white vinegar, and salt. Mix to combine. Taste and adjust seasonings.
Once the soy curls are fully hydrated, drain the excess vegetable broth from the soy curls. Heat up olive oil in a skillet over medium heat. Sauté the soy curls for 5-6 minutes, until the soy curls are browned on the outside and cooked all the way through. Remove from heat and set aside.
To assemble the gyros, slice the tomatoes and red onion. Warm up the pita in a pan or in the oven. Assemble your gyros with lettuce, tomatoes, and red onion. Top with soy curls and drizzle tzatziki over to finish. Serve immediately.