Lightly caramelize the onions by heating the butter in a large skillet over medium heat. Add the onions and cook, stirring occasionally, until lightly caramelized, about 15-20 minutes; season with sea salt to taste, then set aside. Side Note: If the skillet gets dry while the onions cook, add a few tablespoons of water.
Make the burger mixture by combining the ground beef, feta cheese crumbles, parsley, oregano, basil, Worcestershire sauce, garlic, sea salt, and freshly cracked pepper, to taste, in a large bowl and mixing gently with your hands; form into four same-sized patties.
Cook the burgers by heating a grill pan over medium heat and coating it with olive oil cooking spray.
Place the patties on the grill pan and cook for 3-4 minutes on each side or until the desired degree of doneness. Remove from heat.
Build your burgers by layering the bottom of the toasted bun with the burger, lightly caramelized onions, additional feta cheese crumbles, lettuce, tomatoes, some steak sauce, and the other bun. Enjoy.