Season both sides of the duck with the salt, sugar, coriander, paprika, and cinnamon. Add the peppercorns and the seasoned duck to the rendered fat, making sure the duck is submerged. Cover the pan with foil and place and cook for about 4 to 6 hours, until fork tender.
For the quick carrot pickle:
Place a medium pot over medium heat and add the sugar, shallot, garlic, jalapeno, bay leaf, salt, coriander seeds, peppercorns, white wine vinegar, and water. Bring to a boil, stirring until the sugar is dissolved. Add the carrots, and bring back to a simmer. Allow to simmer until the carrots are al dente, about 5 minutes. Remove from heat and set aside to cool.
For the aioli:
Whisk the egg yolks, garlic and lime juice together. Whisk in 1/3 to 1/2 cup liquid duck fat. Add the lime zest and season with hot sauce, and salt, to taste.
Remove the duck from the oven. Carefully remove from the fat and place skin-side down onto a cast iron pan over high heat to crisp the fat, about 2 minutes. Strain the fat and reuse it a couple more times, if desired. Remove the duck from the pan and shred with fork.
Slice the slider rolls in half. Place some of the pickled carrots onto the bottom of the bun. Top the carrots with some of the shredded duck. Spoon some of the aioli and cilantro leaves over top and cover with the slider lids and skewer.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.