In a medium bowl mix together ground beef, lamb, egg, ⅛ cup red onion, steak sauce, ½ tablespoon Greek seasoning, oregano, salt and pepper until well combined. Form into small 2-inch round patties and let sit at room temp 15-30 minutes.
Meanwhile prepare tzatziki sauce: Press shredded cucumber into a fine mesh strainer until liquid is drained. In a small bowl stir together yogurt, cucumber, lemon juice, and remaining red onion and Greek seasoning. Cover and chill until ready to serve.
Preheat gas grill to medium-high heat. Add gyro patties, reduce heat to medium and close lid. Grill 3-5 minutes until bottoms are seared. Rotate and grill about 5 minutes more, or until internal temp reaches 145° F for medium-rare.
Toast buns on grill if desired. Top gyro burger patties with tzatziki sauce and feta cheese, serve immediately.
Use lean ground beef combined with ground lamb. Too much fat can cause flare-ups on the grill and ends up cooking off anyways
Adding an egg to mixture helps to hold all ingredients together and adds moisture
Let prepared gyro patties sit at room temp before cooking. This ensure more even cooking throughout
Start with higher heat on grill then reduce. This creates a sear on outside then slowly cooks meat, making it easier to achieve ideal medium to medium-rare