Mix the beef, onion, parsley, garlic, cumin, allspice, salt and pepper. Split each pita into two semicircles and open the pockets carefully.
Divide the beef among the eight pockets, pressing it into an even layer no thicker than 1/4 inch. Brush both outer faces with olive oil.
Heat two large skillets over medium heat. Cook four pockets at a time across the skillets for 3 to 4 minutes per side, adjusting the heat if the bread darkens quickly; repeat with the remaining pockets.
Stand each pocket on its open meat edge for 1 minute, then check that the center of the beef reaches 160°F; return any underdone pockets to low heat for another minute or two.
Mix the yogurt and lemon juice. Cut the crisp pockets into wedges and serve with the sauce.