Melt 4 tablespoons of the butter in a 10-inch cast-iron skillet over medium heat. Add the onions, garlic, thyme, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the onions are very tender and caramelized, about 15 minutes.
Meanwhile, melt the remaining 2 tablespoons of butter in a small saucepan over medium heat, then brush butter on both sides of the bread. Transfer to a baking sheet. Broil until lightly toasted and golden, about 1 minute per side. Set aside for serving.
Once the onions are caramelized, add the sherry and continue to cook until the liquid has reduced completely, about 1 minute. Pour in the beef broth and Worcestershire and scrape any brown bits from the bottom of the skillet with a wooden spoon. Simmer until the broth has reduced by half, about 2 minutes more. Remove from the heat.
Stir in the cream cheese, 1/2 cup of the mozzarella, sour cream and Parmesan until smooth and creamy. Top with the remaining 1/2 cup mozzarella. Broil until the mozzarella on top is lightly browned and the dip is bubbling at the edges, about 2 minutes. Top with chopped chives and serve with toasted bread.