To make the mayonnaise: In a food processor, combine the egg yolks, salt, mustard, and lemon juice; process together to form a stable base. With the motor running, add a few drops of the oil, the mixture will begin to thicken. Then, begin adding the remaining oil in a thin, steady stream. Scrape down the sides of the processor and pour the mayonnaise into a small bowl. Mix in the black pepper and parsley, cover, and refrigerate; may be kept for up to 3 days.
To prepare the soft-shell crabs: Pour the buttermilk in a shallow bowl large enough to hold all the crabs. Season the buttermilk with crab boil seasoning, salt, and pepper. Lay the crabs in the buttermilk mixture and marinate, covered, in the refrigerator for 1 hour; soaking the crabs in buttermilk helps plump them up when cooked. Remove the crabs from the milk bath and let the excess drip off.
Pour 2 inches of oil in a heavy frying pan or large pot and heat to 350 degrees F.
Put the flour in a pie dish or plate and season with a fair amount of salt and pepper. Dredge the crabs in the seasoned flour to coat, shaking off the excess. Lay the crabs in the hot oil in a single layer without crowding; you may have to fry them in batches. Be careful, the crabs have a tendency to pop and spatter. Cook the crabs for about 3 minutes on each side, turning once, until golden brown. Drain on paper towels.
To build the sandwich: Spread the lemon pepper mayonnaise on both halves of the toasted bread, add the lettuce, tomato, bacon, and avocado. Lay a fried soft-shell crab on top and close up the sandwich.
Contains Raw Eggs: The Food Network Kitchen suggest caution in consuming raw and lightly cooked eggs due to the slight risk of salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly refrigerated, clean grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served, use shell eggs that have been treated to destroy salmonella, by pasteurization or another approved method.