Step 1
In a small skillet, heat the oil, two turns of the pan, over medium. Add the green garlic. Cook, stirring often, until fragrant, about 2 minutes. Remove from heat. Mix in the parsley and mint. Place the beef in a medium bowl; season with salt and pepper. Mix in the grated onion with juices, the Worcestershire, Dijon, and the green garlic mixture.Form into 4 patties (thinner in the centers for even cooking). Heat a cast-iron skillet or griddle over medium-high to high. Cook the patties, turning occasionally, for 8 minutes for beef and 10 minutes for turkey. Top with the cheese during the last minute or two of cooking. Tent the skillet loosely with foil to melt the cheese. In a small bowl, whisk the ketchup, mayonnaise, and balsamic drizzle for the balsamic special sauce. Top the bun bottoms with the cheeseburgers, lettuce, tomato, onion, balsamic special sauce, and bun tops.