Heat oil in a skillet over medium-high heat; sauté onion until soft, 3–4 minutes. Add garlic and cook 30 seconds.
Add the ground beef and cook, breaking it up with a spoon, until it reaches 160°F (71°C), about 6 to 8 minutes.
Stir in oregano, cumin, paprika, salt, and pepper; cook 1–2 minutes to toast spices.
Warm the pitas in a dry pan or oven. For the tzatziki, mix the Greek yogurt with the grated, squeezed-dry cucumber half, lemon juice, dried dill, and a pinch of the listed salt.
Stuff pitas with beef, greens, tomato, cucumber slices, and a generous spoonful of tzatziki.