Heat butter and oil in a large pot over medium heat. Add onions, season with 1/2 teaspoon each salt and pepper, and cook, covered, stirring occasionaly, until very tender, 12 to 15 minutes.
Reduce heat to medium-low and cook, uncovered, stirring occasionally, until onions are golden brown, 20 to 25 minutes.
Sprinkle with flour and cook for 2 minutes; stir in vinegar.
Add wine and simmer for 3 minutes. Add thyme, broth, and 4 cups water and bring to a boil. Reduce heat and simmer for 10 minutes. Spoon soup into bowls and sprinkle with cheese and aditional thyme, if desired.
TIPS & TECHNIQUES.
MAKE-AHEAD TIP: Freeze the soup (without cheese) for up to 3 months. Thaw in the fridge overnight. Reheat, covered, in a saucepan over medium heat. Top with chese and additional thyme before serving, if desired.