Make the dressing: Whisk the mayonnaise, cornichons and their liquid, mustard, hot sauce, and salt and pepper to taste in a bowl.
Prepare the salad: Halve the shrimp lengthwise. Whisk the egg and 2 tablespoons water in a medium bowl. In another bowl, whisk the flour, Cajun seasoning and 2 teaspoons salt.
Heat about 1 inch of vegetable oil in a deep skillet until a deep-fry thermometer registers 350 degrees. (A piece of shrimp will sizzle on contact.) Dip each shrimp in the egg mixture, then dredge in the flour mixture. Working in batches, fry the shrimp until crisp and golden, turning once, about 2 minutes. Remove with a slotted spoon and drain on a paper-towel-lined plate; season with salt.
Toss the romaine, tomato, onion and bell pepper with the dressing in a large bowl; season with salt and pepper. Divide the salad among plates and scatter the shrimp on top.