To prepare aioli: Whisk together mayonnaise, dressing, parsley, Cajun seasoning, lemon juice and lime juice. Cover and refrigerate until needed.
To prepare crawfish cakes: Thaw crawfish. Strain liquid from meat; set liquid aside in a medium bowl. Squeeze as much additional liquid from the crawfish meat as possible; add to the bowl. (You will need 2 tablespoons liquid.).
Beat 1 egg in a large bowl. Add crawfish meat, bell peppers, onion, 1/4 cup flour, bread crumbs, Cajun seasoning, mustard and cilantro; mix gently but thoroughly. Form into 8 cakes, each about 1 inch thick and 2 to 3 inches in diameter.
Divide the remaining 1 cup flour between 2 shallow bowls. Whisk together the remaining 2 eggs and 2 tablespoons liquid from the crawfish in a third bowl.
Coat both sides of each cake in flour, then in egg wash, then in flour from the second bowl.
Heat oil in a medium frying pan over medium-high heat. Add cakes to pan, working in batches if necessary; cook until golden brown on both sides, about 2 - 3 minutes per side.