Preheat your oven to 375°F. Line a rimmed baking sheet with parchment paper.
In a large bowl, combine 1 pound ground beef, 1 pound ground lamb, the grated and drained onion, ½ cup yogurt, 3 tablespoons olive oil, 2 tablespoons paprika, 2 tablespoons cumin, 1 teaspoon coriander, 2 teaspoons oregano, 2 teaspoons salt, 2 teaspoons black pepper, 2 tablespoons lemon juice, and 3 pressed garlic cloves.
Mix everything thoroughly with your hands until the mixture is very well combined and slightly sticky.
Divide the meat mixture evenly among 4 rectangular sheets of parchment paper.
Place another piece of parchment on top of each portion. Using a rolling pin, spread the meat into a very thin, even layer, rolling it all the way to the edges of the parchment. Remove the top sheet.
Starting at the shorter end of the parchment, fold the meat-covered parchment over itself several times, creating folds about 1½ inches wide.
Place the folded parchment packets seam-side down on the prepared baking sheet. Bake at 375°F for 30 to 35 minutes, carefully draining off some of the accumulated fat and juices halfway through cooking.
Remove the baking sheet from the oven and carefully unravel the meat from the parchment. Spread the meat flat directly on the baking sheet.
If you prefer more browning, return the unfolded meat to the oven for 5 to 10 minutes, until lightly crisped around the edges. Remove from the oven and sprinkle with 1 to 2 tablespoons lemon juice.
While the meat cooks, stir together ½ cup yogurt, 2 tablespoons lemon juice, and salt to taste to make the yogurt sauce.
Serve the sliced or torn doner meat in warm pita or lavash with yogurt sauce, chopped parsley, sliced onions and tomatoes, and chili sauce if using.