Start the tzatziki: Grate the cucumber, toss with a pinch of salt, and let sit in a strainer over the sink for 5 minutes. Squeeze it in a clean kitchen towel until no more liquid comes out. Stir into yogurt with garlic, lemon, olive oil, and herbs. Season with salt. Refrigerate while you cook.
Prep produce: Dice tomato, slice red onion, and chop parsley. Grate onion for the meat (do not squeeze).
Brown the beef: Heat skillet over medium-high heat. Add ground beef and grated onion. Cook, breaking apart, until browned and liquid evaporates — about 6 minutes.
Add spices: Push beef to one side. Add a drizzle of oil, then garlic, cumin, oregano, smoked paprika, cinnamon, 1 tsp salt, and pepper. Stir for 30 seconds, then mix into beef. Cook 1 more minute. Taste and adjust salt.
Warm pitas: Heat each pita over a gas flame or in a dry skillet for about 20-30 seconds per side. Stack in a towel to keep warm.
Assemble: Lay a warm pita flat. Spoon tzatziki down the center. Top with beef, tomato, red onion, and parsley. Fold bottom up, then roll. Serve immediately with extra tzatziki on the side.
Let the meat rest: After cooking, let the beef sit in the warm skillet for 2 minutes before assembling to prevent soggy pitas.