Start by carmelizing the onions in the butter over medium heat in a saute pan for about 30 minutes, turning the heat to low when the onions start to brown. Add 2 tablespoons balsamic vinegar to the onions, stirring frequently.
Meanwhile, mix the ground lamb with the salt, pepper, the remaining tablesoon of balsamic vinegar, and minced garlic. Form two patties and make indentations into each patty. Fill each indentation with bits of goat cheese and pinch closed.
Fry the bacon over medium heat and set aside. Fry the patties in the bacon fat, turning only once. Toast the buns and slather with aioli of your choosing. I mixed chopped oven dried tomatoes with mayonnaise.
Pile all the finished ingredients onto the bun. Enjoy with lots of napkins!