In a large stainless steel or cast iron saucepan, 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 the bottom of the pot threaten to burn, add 1 tablespoon of water, scrape up the browned bits, and continue cooking. Season with salt and pepper.
Add sherry and bring to a simmer, scraping up any browned bits. Cook until alcohol smell is mostly gone, about 3 minutes. Add stock and bay leaf, raise heat to medium-high and bring to a simmer. Lower heat and simmer for 20 minutes.
Add cider vinegar and season with salt and pepper.
Preheat broiler and position 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 oven-proof 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 the bowls. Set the remaining 4 toasts in each bowl, 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.
Enjoy hot.