In a large pot, melt butter over medium heat.
Add the onions and cook slowly, stirring occasionally, until they are soft and dark brown.
Stir in the garlic and cook for 1 minute, then remove the skillet from heat.
Deglaze the skillet by adding red wine or ½ cup broth.
Simmer for 2 minutes while scraping up browned bits.
Transfer the contents to a soup pot.
Add the caramelized onions, thyme, bay leaf, and the remaining broth to the soup pot.
Cover the pot and let the soup simmer gently for 20 minutes.
Remove and discard the thyme sprigs and bay leaf.
Season the soup with salt and pepper to taste.
Ladle the soup into bowls and top with a slice of chaffle or low-carb bread.
Sprinkle ¼ cup of cheese over each bowl and place under the broiler until the cheese is melted, hot, and bubbly.