Pour the vinegar into a small saucepan and bring to a boil. Let boil until reduced by half and is thick and syrupy – about 10 minutes. Remove from heat and let cool.

Leaving the root ends attached, cut the Romaine lettuce hearts into quarters vertically. Cut the onion in half vertically and remove the peels. Cut each onion half into 6 vertical slices. Brush the lettuce quarters with olive oil and toss the onion slices with olive oil.

Heat a gas or charcoal grill to medium-high heat. Grill the lettuce quarters about 1-2 minutes on each side. You want them to start to brown but not to get too soft. Remove and transfer to a platter. Grill the onion slices until browned and soft, about 5 minutes. Scatter over the lettuce quarters. Drizzle the balsamic syrup over the lettuce and onions. Drizzle with some olive oil, season with salt and pepper, and top with the crumbled goat cheese.