In 4-quart Dutch oven, melt butter over medium heat. Cook onions in butter 35 minutes, stirring occasionally, until lightly browned and very tender. Add garlic; cook 1 minute. Add 1 cup of the broth, stirring to loosen browned bits from bottom of pan. Stir in 2 cups broth. Heat to boiling; reduce heat. Cover; simmer 10 minutes. Remove from heat; cool 30 minutes.