Step 1Place the duck legs on a rack on a baking sheet and rub them with the salt, bay leaf, thyme, parsley, peppercorns, coriander, mint and sugar. Cover loosely with a sheet of parchment paper and allow to cure for 24 hours in the refrigerator.
Step 2Heat the oven to 250 degrees. In a large heavy-bottomed Dutch oven, sear the duck legs in one tablespoon of canola oil over medium-high heat until you get a bit of color, about 2 minutes. Add the onion and carrot and saute until softened, 7 to 8 minutes. Add the white wine and reduce by half, about half an hour.
Step 3Add the chicken stock and braise the duck legs in the oven, covered, until the meat is tender and falling off the bone, about 3 1/2 to 4 hours. (If it begins to bubble, turn down the heat.) Allow the meat to cool, then remove from the braising liquid; the braising liquid can be reserved for another use such as for a soup base.
Step 4Remove the meat from the bone and place it in a bowl. Place the bowl of duck meat on top of a bowl of ice.
Step 5In a small pan, heat the duck fat over medium-low heat until it's melted. Slowly pour the duck fat over the duck meat, using a fork to emulsify the duck meat with the duck fat until fluffy and smooth. Add the Dijon mustard and adjust seasoning to taste. Transfer to a serving dish or container; the restaurant serves rillettes in a French canning jar.