Melt 2 tablespoons butter in a large skillet over medium heat. Add sliced onions; cook, stirring occasionally, until just beginning to soften, about 5 minutes.
Add ¾ cup broth, the minced garlic and ½ teaspoon thyme; cook, stirring occasionally, until the onions are deeply browned and the broth is fully absorbed, 10 to 15 minutes.
Remove from heat and stir in 1 tablespoon vinegar; let cool for 10 minutes.
In a food processor, combine the remaining ½ cup softened butter, ½ teaspoon salt, ½ teaspoon Worcestershire, ¼ teaspoon pepper and half of the caramelized onions.
Process until the mixture is mostly smooth, stopping often to scrape down the sides of the bowl as needed, 1 to 2 minutes.
Add the remaining caramelized onions; pulse until combined and coarsely chopped, 8 to 10 pulses, stopping to scrape down the sides of the bowl as needed.
Transfer to a small serving bowl.
