Put lamb, soy sauce, and onion in a bowl and season with salt and pepper to taste. Mix with hands to combine. Form lamb mixture into 4 patties.
Heat avocado oil in a skillet over medium heat. Cook lamb patties until browned, 5 to 8 minutes on each side. An instant-read meat thermometer, inserted into the center of a patty, should read at least 145 degrees F (65 degrees C). Sprinkle with feta cheese and serve.