Slowly Soften the Onions: In a heavy-bottomed pot, melt the butter over medium heat until it begins to foam. Add the sliced onions, stirring well to coat them evenly with the butter. Lower the heat to medium-low and cook the onions gently for about 30 minutes. Stir occasionally to ensure they soften evenly and become translucent.
Deepen the Onion Flavor: Turn the heat up to medium or medium-high and sprinkle in a pinch of salt. Continue cooking for another 20 to 30 minutes, stirring frequently. This step will caramelize the onions slowly, turning them golden brown and releasing their natural sweetness and rich flavor.
Add Wine to Deglaze: Pour in the white wine to deglaze the pot, scraping the bottom to lift any browned bits. Let the wine simmer and reduce for about 2 minutes until mostly evaporated, enriching the base of your soup.
Incorporate Flour: Sprinkle the flour evenly over the caramelized onions and stir continuously for 1 minute. This will help thicken the soup gently without clumping.
Add Broth and Herbs: Pour in the broth, then add the bay leaves and thyme sprigs. Bring the mixture to a gentle simmer. Cover the pot, reduce heat to low, and allow it to simmer softly for 30 minutes to marry the flavors.
Prepare the Cheesy Toast: While the soup simmers, preheat your grill or broiler on high. Toast the bread slices on both sides until lightly golden. Place cheese slices on the toasted bread, then return under the grill until the cheese melts fully and develops golden spots.
Final Seasoning and Serving: Remove the bay leaves and thyme sprigs from the soup. Taste and adjust seasoning with salt and pepper as needed. Ladle the hot soup into bowls, then float one or two cheesy toast slices on top. Garnish with fresh parsley or thyme if desired, and serve immediately.