For the oxtail: Sprinkle the oxtail with salt and pepper on all sides. Sear the oxtail in the olive oil in a pressure cooker on the saute function until golden brown on all sides. Add the carrot, celery and onion and cook until starting to caramelize. Add the thyme, tarragon, rosemary, chicken stock, red wine and veal demi-glace. Close the pressure cooker according to the manufacturer's instructions and pressure cook for 50 minutes. Release the steam according to the manufacturer's instructions and remove the oxtail to a plate to cool, then remove the meat from the bones. Strain the liquid from the pressure cooker into a medium saucepan and reduce the sauce until slightly thickened. Add the butter and blend the sauce with an immersion blender until it looks glossy. Taste and adjust the seasoning.
For the aioli mayonnaise: Mix the mayonnaise, vinegar and garlic together in a small bowl. Set aside.
For the caramelized onions: Warm a large saute pan over medium heat and add the olive oil. Once hot, add the onions followed by the sugar, salt and pepper. Reduce the heat and cook slowly, stirring occasionally, until all of the onion is caramelized, 15 to 30 minutes.
For the burger: Preheat the oven to 350 degrees F.
Combine the ground beef, Worcestershire sauce, rosemary, tarragon and salt in a large bowl. Reserve 1/2 cup oxtail for garnishing the burgers, then add the rest of the oxtail to the bowl. Mix well, but don't overmix. Form six 1/4-pound burger patties. Sprinkle both sides of each patty generously with salt and pepper.
Set a large cast-iron skillet over medium-high heat and add the canola oil. When the oil shimmers, add the patties (you may need to do this in batches). Cook the patties all the way through, about 4 minutes, then flip and cook another 2 minutes. Add the butter, thyme sprigs and garlic to the skillet and let the butter get foamy, about 30 seconds, then baste the patties, about 1 minute. Flip the patties and baste the other side, 1 minute more. Remove the skillet from the heat and top each patty with some cheese. Place the skillet in the oven to melt the cheese, around 2 minutes.
In the meantime, season the arugula with lemon juice, salt and pepper in a bowl.
Spread a bottom bun with aioli mayonnaise and top with a heaping spoonful of the caramelized onions, some chopped chives, a patty, some reserved oxtail and a spoonful of oxtail gravy. Finish with a handful of dressed arugula and add the top bun. Repeat with the remaining ingredients to make 5 more.