Preheat oven to 375°F. Line two sheet pans with parchment paper.
In a large bowl, combine ground beef, puréed onion, minced garlic, olive oil (if using), oregano, paprika, cumin, salt, and black pepper. Mix just until combined.
Divide the meat mixture into four portions. Place one portion on a sheet of parchment paper and top with another sheet.
Using a rolling pin, roll the meat out thin (about ⅛–¼ inch thick). Remove the top parchment.
Starting at one end, loosely roll the bottom parchment with the meat inside to form a log. Transfer to the prepared sheet pan. Repeat with remaining portions.
Add sliced red onion and tomato to the pan. Drizzle lightly with olive oil and season with salt and pepper.
Bake for 18–22 minutes, until the meat is fully cooked and no longer pink (160°F internally).
Remove from oven, unroll the parchment, and tear the meat into rough pieces. Spread evenly across the pan.
Broil on high for 3 minutes, watching closely, until the edges are lightly browned and crisp.
Serve warm in pita with tzatziki, roasted vegetables, and fresh cucumber.