Place onion in a food processor and blend until finely chopped. Transfer onion to a piece of cheese cloth and squeeze out the liquid. Place onion in a large bowl.
Mix lamb, beef, garlic, oregano, cumin, marjoram, thyme, rosemary, black pepper, and salt with the onion using your hands until well mixed. Cover bowl with plastic wrap and refrigerate until flavors blend, about 2 hours.
Preheat oven to 325 degrees F (165 degrees C).
Place meat mixture in a food processor and pulse until finely chopped and tacky, about 1 minute. Pack meat mixture into a 7×4-inch loaf pan, ensuring there are no air pockets. Place loaf pan into a roasting pan and pour enough boiling water around the loaf pan to reach halfway up the sides.
Bake in the preheated oven until no longer pink in the center, 45 to 60 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Pour off any accumulated fat and cool slightly.
Thinly slice the gyro meat mixture.
Spread 1 tablespoon hummus onto each pita bread; top each with gyro meat mixture, lettuce, tomato, red onion, feta cheese, and tzatziki sauce.
I use homemade pita bread, hummus, and tzatziki sauce.
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