In a large nonstick skillet sauté onions with butter over medium heat, tossing to coat. Cook until brown and soft, about 10 minutes.
In a bowl, whisk together eggs, chicken broth and balsamic vinegar.
Divide onions among 4 ramekins; pour egg mixture on top. Cover each tightly with plastic wrap, then foil. Place ramekins in a steamer set over a large pot of water at a low boil. Cover and steam for 12 to 15 minutes, until custard is set but not firm.