Preheat the oven to 300°F (150°C) and set an oven rack in the middle position. Line a baking sheet with heavy-duty aluminum foil.

Place the beef, lamb, bread crumbs, salt, pepper, oregano, onion powder, garlic powder, and cumin in the bowl of a food processor fitted with the steel blade. Process until the mixture is well combined and paste-like, scraping down the bowl as necessary, about 1 minute.

Using moist hands, form the mixture into two free-form loaves about 1 in high, 8 in long, and 4 in wide directly on the prepared baking sheet. Bake for 40 to 45 minutes, or until meat is firm in the center and reaches an internal temperature of 160°F (71°C). Let sit for about 10 minutes, or chill until ready to slice, up to 4 days. Transfer the loaves to a cutting board and slice between ⅛ and ¼ inch thick.

To serve, working in batches, heat about 1 tablespoon of oil in a nonstick skillet over medium heat. Cook the gyro slices in a single layer until lightly browned, 30 to 60 seconds per side. Serve with pita bread, tzatziki sauce, tomatoes, onion, lettuce, and feta.