Step 1

In a large stainless steel saucepan, or in 2 large stainless steel or cast iron skillets, melt butter over medium-high heat until foaming. Add onions and cook, stirring occasionally, until softened, about 8 minutes. Lower heat to medium-low and cook, stirring frequently, until onions are very sweet and a rich golden-brown color, 1 to 2 hours. If browned onion juices on bottom of pot threaten to burn, add 1 tablespoon (15ml) water, scrape up browned bits, and continue cooking. Season with salt and pepper.

Step 2

Add sherry and bring to a simmer, scraping up any browned bits. (If using 2 skillets, divide sherry between them, then scrape onions and liquid from both pans into a pot or Dutch oven to continue.) Cook until alcohol smell is mostly gone, about 3 minutes. Add stock, thyme, and bay leaf; raise heat to medium-high; and bring to a simmer. Lower heat and simmer for 20 minutes.

Step 3

Add fish sauce, if using, and cider vinegar. Season with salt and pepper. Discard thyme sprigs and bay leaf.

Step 4

Preheat broiler and adjust oven rack to top position. Butter toasts and rub with garlic clove until fragrant. Spoon a small amount of broth into the bottoms of 4 ovenproof serving bowls, then top with half the toasts. Sprinkle some grated cheese on top of toasts, then spoon more soup and onions on top, nearly filling bowls. Set remaining 4 toasts in bowls, pushing to nearly submerge them. Top with remaining grated cheese and set bowls on a rimmed baking sheet. Broil until cheese is melted and browned in spots. Garnish with chives and serve.