Add the butter to a large pot over medium heat. Add the onions and sauté, stirring frequently, until they're caramelized, about 20-30 minutes.
Add the garlic and cook for another 1 minute. Pour in the wine and let it simmer for 2 minutes.
Stir in the beef stock, bay leaf, and thyme. Simmer for about 10-15 minutes.
While the soup is cooking, make your chaffles. Mix the mozzarella, cheddar, and eggs in a bowl. Spoon out enough batter to just fill a mini waffle maker. Cook until the chaffle is golden brown and the waffle maker's indicator light goes off or steam stops, typically about 3-4 minutes. Repeat with the remaining batter, which should make about 6 small chaffles.
Fish out the bay leaf and thyme from the soup and discard them. Ladle your soup into oven-safe dishes. Place a chaffle on top of each, then sprinkle some shredded Gruyère cheese on top of that. Broil until the cheese is melted, about 2-3 minutes.