Preheat oven to 450ºF. Spray 2 heavy large baking sheets with vegetable oil spray.
Arrange onions in single layer on prepared sheets. Dot onions with 4 tablespoons butter, dividing equally. Season with salt and pepper. Bake until onions are dark brown on bottom, about 35 minutes.
Meanwhile, boil broth in heavy large skillet over high heat until reduced to 1/2 cup, about 6 minutes. Add sugar and vinegar and whisk until sugar dissolves and moisture comes to boil.
Add onions to sauce; reduce heat to medium-low. Simmer until liquid is slightly reduced, about 5 minutes. Season with salt and pepper. (Can be prepared 1 day ahead. Cover and refrigerate. Reheat over low heat before continuing).
Cook green beans in large pot of boiling salted water until crisp-tender, about 5 minutes. Drain well. Return beans to same pot. Add remaining 2 tablespoons butter and toss to coat. Mound beans in large shallow bowl. Top with onion mixture and serve.