Melt 2 tablespoons of the butter in the skillet. Add the sliced onions and cook over moderately high heat, stirring, until soft and caramelized, about 20 minutes. Add the garlic and cook until softened, 2 to 3 minutes. Add the Chicken Stock and thyme and boil until reduced by half, about 17 minutes. Add the balsamic vinegar and simmer for 5 minutes. Strain the sauce into a medium saucepan, pressing on the solids. Boil the sauce until reduced to 1 cup.