Put the ground beef and lamb in a large bowl and knead until nicely combined (we used just lamb so skipped this).

Put the onions and parsley in a food processor and pulse to create a coarse paste. Wrap the mixture in a clean kitchen towel (no fabric softener, please!), twist the towel like a candy wrapper, and squeeze out as much liquid as you can. Add the onion mixture (and pine nuts if using) to the ground meat. Add the salt and pepper (and za'atar if using) and knead again to blend thoroughly.

To make the tahini sauce, mix together all ingredients in a medium bowl and add enough water to get a thick but runny texture. Or process in a mini food processor.

Stuff each pita half with about 1/2 cup (125 g) of the meat mixture. Gently press the pita halves between your palms to distribute the stuffing evenly. Make sure it comes all the way to the cut edge of the pita so that the meat can be seared by the grill fire.

Generously brush both sides of each pita with the oil or beef fat (if you're using beef fat, melt it gently in a small pan until liquid first.

To grill the pita: Make sure the grill is clean and heat it to high.

Place the stuffed pita halves on the grilled and sear until they turn golden brown and have nice grill marks, about 1-2 minutes on each side. Cut each half into quarters (we skipped this part by mistake), and sear for 2 minutes on each of the sides with the filling exposed. Move the quarters to a cooler spot on the grill or turn down the heat, lay them flat again, and continue cooking another 2 to 3 minutes per side depending on your grill heat. By this time, the pitas will have absorbed the fat from the filling and will be juicy inside and crunchy and deep golden brown on the outside, and the meat filling will be perfectly cooked. Total grilling time will be about 12 minutes.

If you're cooking indoors: Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper. Heat a large cast-iron skillet over medium-high heat for 3 to 4 minutes to get it really hot. Working in batches, if needed, lay the stuffed pitas flat on the hot skillet in one layer. Sear them until they turn deep golden brown on the first side, 2 to 3 minutes. Using tongs, flip the pitas and sear for 2 to 3 minutes on the second side. Flip the pitas once again so the cut side (with the expsed filling) is now facing the skillet and sear for 2 to 3 minutes, until the filling is set and nicely browned. Transfer to a large plate and repeat to cook the remaining pitas. Let cool for a coup;e of minutes so you can handle them, then cut each pita in half. You should now have 24 stuffed pita quarters. Arrange the stuffed pita quarters on the prepared baking sheet and bake for 10 minutes.

Serve at once. (With tahini sauce or another dipping sauce oif you like.)