Preheat oven to 400F.
In a large bowl, combine the beef, onion, garlic, Greek yogurt, spices, and tomato paste. Mix thoroughly for 3–4 minutes until the mixture is well combined. You can also do this in a food processor by pulsing everything for 30–60 seconds. Divide into 4 portions.
Lay out a large sheet of parchment paper. Spread the ¼ of the meat mixture into the center. Place another sheet of parchment paper right over. Use a rolling pin to roll out the meat between the two parchment papers into a thin rectangle, about ⅛ inch (3 mm) thick.
Starting from a long edge, roll the parchment paper and meat into a tight log. Repeat with the remaining meat.
Place the wrapped logs on a baking sheet. Bake for 20-25 minutes, or until the center reaches 160F.
Let the meat rest for 5–10 minutes. Unwrap and tear into very thin strips. If the meat did not pick up any crispy edges, place it back onto the baking sheet and broil for 1-2 minutes until the edges are slightly browned and crispy.
While the meat cooks, make the tzatziki. Grate the English cucumber. Squeeze out as much liquid as possible using a clean towel or paper towels. Combine in a bowl with Greek yogurt, garlic, dill, lemon juice, and salt. Taste and add more salt if needed. Drizzle with olive oil.
Serve in warm lavash with tomatoes, sumac onions, and tzatziki sauce.