If there’s a lot of fat in the pot, pour it off until you have 2 tablespoons left. Add carrot, onion, leek (if using) and cook on medium-high heat until lightly browned at edges, about 5 to 6 minutes. Season with salt and pepper. Add tomato paste and cook for 2 minutes; it will get a little darker. Add the red wine and cook until it has reduced to just a puddle, about 3 minutes. Return the short ribs and any juices that have collected to the pot. Add garlic and thyme sprigs, then pour broth over ribs and vegetables. Cover with lid, transfer to the oven, and braise until the short ribs are falling off the bone and everyone in your home is falling over from how good it smells, about 2 1/2 to 3 hours.
Meanwhile, prepare the onions: [If you have another large Dutch oven (fancy!) you can use it here. A large soup or stock pot will do, too. Or, you can use a large, deep frying pan for just the onions and finish the soup in the short rib’s pot later.]
Melt butter over medium heat. Add the onions, toss to coat them in butter and cover the pot. Reduce the heat to medium-low and let them slowly steep for 15 minutes. They don’t need your attention.
Uncover the pot, raise the heat slightly and stir in salt — I start with 2 to 3 teaspoons of kosher salt. Cook onions, stirring every 5 minutes (you might be fine checking in less often in the beginning, until the point when the water in the onions has cooked off) for about 40 to 90 minutes longer. Onions are caramelized when they’re an even, deep golden brown, sweet and tender. Add sherry and scrape up any onions stuck to the pan, then simmer it until it disappears and you’re swatting strangers who crept in, enticed by the aroma, out of your kitchen.
Finish the short ribs: When the short ribs are cooked, use a slotted spoon to remove them from the broth and transfer to a plate to cool slightly. Strain the broth, discarding the vegetables. If the broth looks fattier than you prefer, you can use a fat separator to remove it, or carefully spoon it off the surface. Discard the short rib bones and pull the meat into large bite-sized chunks. You can de-fat the ribs a bit here, too, if there are easily-removed pieces.
Place the caramelized onions in the final soup pot, if they’re not already there, and rewarm over medium-high. Add broth and bring it to a simmer and season to taste with more salt and pepper. Add short ribs to broth and gently simmer everything together for 10 to 15 minutes.
To make the cheese toasts: Heat your oven’s broiler (or turn it to its top temperature). Coat a large baking sheet with foil, for easiest cleanup. Gently toast the bread until semi-firm and dry to the touch. Rub each with the raw garlic clove. Divide cheese between the toasts and return the tray to the oven until the cheese has melted and the toasts are browned on top.
To serve: Ladle ribs and broth into bowls and sink a cheese toast halfway in. Sprinkle with chives.
Do ahead: I love making this a day ahead of time; short ribs are fantastic the second day, even better, you could argue, plus any excess fat in the broth will be easier to remove once chilled. You can make everything but the cheese toasts early, or just a component or two (the short rib braise, the caramelized onions). Rewarm over medium-high heat until simmering to serve.