Pre-heat a griddle pan to medium-high heat.
In a food processor, combine the oregano, red pepper, parsley, mint, shallot, garlic, vinegar, olive oil and lemon zest and pulse into a fine chimichurri sauce. Place the meat in a bowl and season with salt and pepper. Combine with the herb mixture and form patties.
Combine the feta sauce ingredients in a clean food processor bowl and process into a creamy sauce.
Cook the burgers to desired doneness, 7-8 minutes for pink center on lamb or beef, 10-12 minutes for fully cooked chicken. Serve on toasted rolls or bread with lettuce, tomato, onion and sauce.