Heat the oven to 400°F (200°C) with a rack in the upper-middle position. Squeeze the grated onion firmly in a clean towel, then combine the dry onion, beef, garlic, oregano, cumin, coriander, paprika, salt, and pepper until the seasonings are evenly dispersed and the mixture looks cohesive. Do not add the onion liquid.

Divide the beef in half. Put one portion between two sheets of parchment and roll it into a roughly 10-by-12-inch rectangle, about 1/8 inch thick. Peel away the top parchment, fold the lower parchment and meat into thirds, then set the packet seam-side down on the sheet pan. Repeat with the remaining portion.

Bake for 15 minutes. Open one packet carefully away from your face. Check the thickest folded area with an instant-read thermometer; the ground beef must reach 160°F (71°C). If it has not, close the packet and continue in 2-minute increments.

Let the packets stand for 2 minutes. Unwrap them carefully, letting steam escape away from you. Tear the beef into irregular bite-size pieces, spread in one layer on the sheet pan, and broil 2 to 4 minutes, watching constantly, until the thin edges are browned and crisp.

Stir the yogurt, garlic, lemon juice, olive oil, and salt until smooth. Fill each warm pita with one-quarter of the beef, sauce, and any optional vegetables. Serve immediately.