Preheat oven to 450°F.
Spray or oil baking sheet. Arrange onions in a single layer. Dot onions with 4 Tbs. butter, dividing equally. Season with salt and pepper. Bake until the onions are dark brown on bottom, about 35 minutes.
Meanwhile, boil the broth over high heat until reduced to ½ cup. Add sugar and vinegar and whisk until the sugar dissolves and mixture boils.
Add onions to sauce, reduce heat to medium-low. Simmer until liquid is slightly reduced, about 5 minutes. Season with salt and pepper if necessary.
Steam beans to taste, el dente, toss with 2 tbs. butter and pile onions on top to serve.