Preheat oven to 300 degrees F (150 degrees C). Grease a large baking pan with sides.
Place beef, lamb, flour, salt, oregano, dry mustard, 2 1/2 teaspoons garlic powder, black pepper, cayenne pepper, chile pepper, paprika, and Italian seasoning in a bowl; knead all ingredients together with your hands until mixture is like a paste, about 10 minutes. Roll the meat mixture into a large loaf shape, and place into the prepared pan.
Bake in the preheated oven until no longer pink in the center, about 2 hours. An instant-read thermometer inserted into the center should read at least 160 degrees F (70 degrees C). Remove from oven; place meat on a drip rack until completely cool to allow juices and fat to drain. Refrigerate meat for 6 hours or overnight.
Stir evaporated milk, sugar, and 2 teaspoons garlic powder together in a small bowl; gradually stir in vinegar just until mixture thickens.
Slice donair meat into strips about 1/8-inch to 1/4-inch thick. Fry meat strips in single-sandwich batches in a skillet over medium-high heat until meat begins to crisp at the edges; remove from pan. Quickly rub 1 pita round lightly with water; fry pita in same frying pan, turning once, just until warm, about 1 minute.
Spread pita with about 1/2 tablespoons of sauce mixture, top with donair meat, and garnish with onion, tomato, and more sauce; repeat with remaining meat and pita.
Beef can be used in place of lamb if desired.
Flour acts as a binder, but it is possible to make the recipe without using flour.
In place of kneading meat mixture in Step 2, you can pass mixture through a grinder until finely ground. The finished product should be similar to processed meat, so that it can be thinly sliced and then pan fried without crumbling.
You can also top with mozzarella cheese if you like.
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