Braise the short ribs: Preheat the oven to 325 degrees F. Toss the short ribs with the flour and season liberally with salt and pepper. Heat the canola oil in a large Dutch oven over medium-high heat until it begins to shimmer. Add the short ribs and cook until browned on all sides, about 8 minutes. Transfer to a plate.
Add the carrot, chopped onion and leek to the pot and cook, stirring, until lightly browned, about 4 minutes. Return the short ribs to the pot and stir in the thyme sprigs, bay leaf, garlic and tomato paste. Add the wine and bring to a simmer, then cook until reduced by half, about 5 minutes. Add the stock and return to a simmer. Cover, transfer to the oven and braise until the short ribs are falling off the bone, 2 hours, 30 minutes to 3 hours.
Meanwhile, caramelize the onions: Melt the butter in a large pot over medium heat. Add the sliced onions, reduce the heat to low and season with salt and pepper. Cook, stirring frequently, until deep golden brown, about 2 hours, 30 minutes. (Add a splash of water and scrape up any browned bits from the pot, if necessary.)
When the short ribs are done, remove from the broth with a slotted spoon, transfer to a plate and let cool slightly. Strain the broth, discarding the solids. Shred the meat, discarding the bones, and toss with 1 cup of the broth. Reserve the remaining broth.
Add the thyme leaves and sherry to the caramelized onions. Cook until the liquid is reduced by half, about 3 minutes; add the reserved broth and cook 30 more minutes. Season with salt and pepper. Skim the fat from the top of the soup. Stir in the vinegar.
Preheat the oven to 425 degrees F. Divide the soup among 6 ovenproof bowls; spoon about 1/3 cup shredded meat into the center of each, then top each with 2 slices toast, trimming the bread if necessary. Top each with 2 slices cheese, letting the cheese hang over the side of the bowl. Transfer to the oven and bake until golden and bubbly, about 10 minutes. Top with chives.
Photograph by Christina Holmes
Courtesy of Food Network Magazine