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1
Heat olive oil and butter in a skillet over low heat. Add onion and cook until starting to soften, 5 to 7 minutes. Add 5 peeled garlic cloves and cook until very soft and browned, breaking them up with the spoon, 5 to 7 minutes or more.
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2
Combine water, oxtail, carrots, onion skin, garlic skin, and bay leaves in a saucepan with a lid. Cover and cook stock over low heat until extremely tender, about 2 hours total.
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3
While oxtail cooks, combine rosemary, sea salt, 2 cloves garlic, black pepper, coriander, and paprika in a mortar. Grind together. Transfer mixture to a dry frying pan over low heat. Roast, stirring occasionally, about 5 minutes. Return mixture to the mortar and crush again. Return mixture to the frying pan and roast over low heat, another 5 minutes. Grind again in the mortar. Repeat this process as needed until the mixture has the consistency of a fine, dry salt. Place rosemary salt in an airtight container until needed.
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4
Remove bones from oxtail and reserve. Chop meat finely and set aside. Strain oxtail stock through a mesh sieve and return it to the saucepan. Discard vegetable remainders. Return oxtail bones to the stock and bring it to a boil over high heat until reduced to about 1 cup. Remove and discard bones. Add fried onion and garlic mixture and return chopped oxtail meat to the saucepan. Reduce heat and simmer until oxtail mixture is sticky and almost dry, about 30 minutes more.
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5
Spread oxtail mixture onto a large plate or baking tray. Mash using a potato masher to be sure there are not lumps. Allow the mixture to cool completely, about 30 minutes.
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6
Combine cooled oxtail mixture, ground beef, bread crumbs, mustard, ketchup, hot pepper sauce, and olive oil in a large bowl. Mix well. Form into 8 burgers and place on a plate. Cover and refrigerate at least 30 minutes.
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7
Immediately before cooking burgers, rub a generous amount of rosemary salt onto each side to form a dry crust.
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8
Preheat an outdoor grill for medium heat and lightly oil the grate. Grill burgers until blackened on the outside and cooked to desired doneness, 5 to 7 minutes.
