Set a medium saucepan over medium heat and add butter. When the butter is melted, add the sliced onions.
Saute for 20 minutes, stirring frequently, until onions are golden brown.
Sprinkle flour over the caramelized onions. Stir to coat the onions in flour and continue cooking for 1 minute.
Gradually add the chicken broth, Worcestershire sauce, thyme, salt, and black pepper. Bring just to a boil, then reduce the heat to a low simmer. Simmer for 10-15 minutes, giving it a good stir occasionally, until thickened.
Adjust seasoning to taste and serve hot.