Place the sliced beef into a flat, oven safe dish. Stir the red wine vinegar, 100ml olive oil, lemon juice, allspice, cinnamon, cardamom, black pepper, salt, tomatoes and garlic together in a bowl until well blended. Pour over the beef, turning slices to coat evenly. Cover and chill for 4 hours.
Preheat the oven to 220 C / Gas mark 7.
Roast the meat until it is no longer pink, about 40 minutes. Cool slightly.
Meanwhile, make the parsley sauce by mixing the parsley, onion, 5 tablespoons olive oil and 3 tablespoons lemon juice together in a bowl. Set tomatoes aside until needed.
Make the tahini sauce by mixing the garlic, tahini, remaining lemon juice and water together in a bowl. Season to taste with salt.
To serve, place the pitta on serving plates. Spoon some of the meat mixture down the centre of each pitta. Top with the parsley mixture, tomatoes and tahini sauce. Roll up the sides of the pitta around the filling and serve.