How to Make It
In a blender, combine the vinegar, mustards, ketchup, paprika, cayenne, garlic, hot sauce, salt, scallions, celery, parsley and eggs on high speed until smooth. With the machine on, add the oil in a thin steady stream until the mixture is thick. Transfer to a bowl and refrigerate until slightly chilled.
In a large pot, combine the salt, cayenne peppers, peppercorns, lemons, garlic, bay leaves, onions and 2 gallons of water and bring to a boil over moderately high heat; cook for 15 minutes. Add the shrimp and cook until just pink, about 2 minutes. Using a slotted spoon, transfer the shrimp to a large bowl. Add the ice to the bowl and strain 4 cups of the hot cooking liquid over the shrimp. Let soak for 5 minutes, then drain.
In a large bowl, toss the cabbage, carrot and red bell pepper with some of the remoulade sauce. Mound on a platter, arrange the shrimp on top and serve the remaining sauce on the side.