Position two oven racks in the upper and lower thirds of the oven and heat the oven to 450ºF.
Heat the olive oil in a 10-inch skillet over medium heat until just shimmering. Cook the onion and garlic in the oil with a pinch of salt, stirring occasionally, until softened, 3 to 5 minutes. Add 3/4 tsp. of the toasted cumin seeds, the allspice, red pepper flakes, and black pepper and cook, stirring, until just fragrant, about 30 seconds. Add the tomatoes and cook, stirring occasionally, until all the liquid has evaporated, 5 to 8 minutes. Remove from the heat.
Meanwhile, pull the lamb meat from the bones and cut it into chunks (you should have about 1-1/2 cups). Pulse the lamb in the food processor until it is roughly chopped, resembling the texture of very coarsely ground meat. Add the tomato sauce, lamb sauce, 1 Tbs. of the mint, the lemon juice, and 3/4 tsp. salt and pulse just to combine. Adjust the salt, pepper, and lemon juice to taste.
Put the pita halves on two baking sheets and brush them generously on both sides with olive oil. Position them with the inner sides facing up and divide the lamb mixture among the pitas, spreading it evenly over the surfaces. Bake until crisp, 5 to 10 minutes. Serve the pitas drizzled with some of the yogurt and sprinkled with the remaining toasted cumin, mint, and some freshly ground black pepper. Serve with the remaining yogurt on the side.