To make the Souvlaki: Place Mozzarella cheese in a medium bowl. In a separate bowl, combine parsley, soy sauce, oregano, salt, and pepper. Pour parsley mixture over cheese, mix well, and refrigerate, covered, for 2 hours.
Arrange Mozzarella cheese skewers, 3 pieces per skewer, and refrigerate.
To make the Burrata-Ganoush: Preheat a griddle over medium-high heat. Cook eggplants on griddle, turning often, until eggplants are very soft, charred on all sides, and have lost about half their original size and collapsed. Remove to a bowl, cover, and allow to cool to room temperature.
When cool enough to handle, remove charred skin from eggplants. In the bowl of a food processor, combine eggplant flesh, tahini, olive oil, lemon juice, and garlic and process to a chunky puree. Transfer to a bowl and fold in stracciatella cheese. Add salt and pepper to taste and set aside.
To make the Batter: In a medium bowl, whisk all-purpose flour, rice flour, za’atar, baking soda, salt, and pepper. Add sparkling water and lemon juice, whisking until just combined (a few lumps is okay).
To make the Salad: In a large bowl, combine arugula, fennel, olives, onion, and radishes. Add oil, lemon juice, and salt and toss gently until evenly coated. Set aside.
To assemble and serve: Preheat a griddle over low heat. Meanwhile, add vegetable oil to a medium saucepan, enough to come halfway up the sides, and heat to 350°F. Line a platter with paper towels.
Lightly dust each souvlaki skewer with rice flour, then coat with batter and fry, holding skewer in oil a few seconds to allow a crust to form before releasing. Cook until golden, 2 to 3 minutes. Transfer to the prepared platter.
Heat lavash on the griddle until soft and warm. Arrange lavash on serving plates and spread burrata-ganoush on top. Top with salad and souvlaki.
