Preheat oven to 325°. Process onion in a food processor until finely chopped, 10-15 seconds. Transfer to a clean kitchen towel; squeeze out as much liquid as possible. Return onion to food processor. Add beef, lamb, garlic, salt, oregano, thyme, rosemary, cumin and pepper. Pulse until mixed, 2-3 times; scrape down sides. Process until meat mixture is sticky and paste-like, about 1 minute.
Transfer meat mixture into an ungreased 9×5-in. loaf pan. Press and smooth top of mixture with a rubber spatula, packing tightly into pan.
Place loaf pan in a larger baking pan; add 1 in. of hot water to larger pan. Bake until a thermometer reads 165°, 60-70 minutes. Remove loaf pan from water bath immediately to a wire rack. Place a second 9×5-in. loaf pan, bottom side down, onto the meat mixture in the pan. Weigh down the top pan by adding several cans to the top pan. Cool like this for 30 minutes.
Remove weighted pan; drain grease from loaf. Return loaf to pan. Cover and refrigerate until chilled, about 3 hours or overnight. Remove from pan, cut widthwise into 1/4-in. slices.
In a large skillet, heat 1 tablespoon oil over medium-high heat. In batches, cook and stir meat slices until lightly browned, 3-5 minutes, adding more oil as needed. Serve in pitas with toppings as desired.
