In a large skillet, melt the butter over medium low heat. Add the onions to the pan.
Cook, stirring occasionally, until golden brown and caramelized, about 30-45 minutes. Adjust heat between medium and low as needed to caramelize the onions without burning.
Stir in the garlic and cook until fragrant, 30 seconds to 1 minute.
Add the beef broth and stir to deglaze the pan. Stir in the Worcestershire sauce, salt, and pepper.
Cook for another 1-2 minutes until the liquid has evaporated. Remove the pan from heat and set aside to cool.
In a medium bowl, combine the sour cream, mayonnaise, and chives.
Once the onions are cool to touch, fold them into the sour cream mixture.
Adjust seasonings to taste, cover, and refrigerate for at least 2 hours to overnight.
Serve with vegetables, chips, and/or crackers.