Mix all gyro seasoning spices together in a small bowl. Set aside.
Mix lemon juice, balsamic vinegar, and ketchup together in a small bowl. Set aside.
Drain and rinse jackruit. Chop and remove any hard parts or parts of the core. Preheat oven to 400F.
Sauté minced garlic in olive oil or water until fragrant, about 2-3 minutes. Add jackfruit and sauté for a few minutes, then add gyro seasoning spices and stir to coat jackfruit in spices. Sauté for 7-10 minutes, until tender, and then shred into small pieces using 2 forks.
Pour lemon juice, vinegar, and ketchup mixture over the gyro meat and stir to coat. Simmer for 5-8 more minutes. Add extra smoked paprika if you want a smokier flavor.
Line a baking sheet with parchment paper as you wait for jackfruit to cool. Bake jackfruit for 15 minutes for a meatier texture.
Slice red onion, cucumber, and tomatoes. Assemble sandwiches by placing gyro meat, tomato slices, cucumber slices, and red onion slices on a wrap. Top generously with tzatziki. Enjoy!