Heat oil in a skillet over medium-high heat; sauté onion until soft, 3–4 minutes. Add garlic and cook 30 seconds.
Add ground beef, breaking it up with a spoon; cook until no pink remains, about 6–8 minutes.
Stir in oregano, cumin, paprika, salt, and pepper; cook 1–2 minutes to toast spices.
Warm pitas in a dry pan or oven. Mix 1 cup plain yogurt with 1/2 grated cucumber, 1 tsp lemon juice, salt, and a pinch of dill for tzatziki.
Stuff pitas with beef, greens, tomato, cucumber slices, and a generous spoonful of tzatziki.