In a small bowl, stir together ½ cup mayonnaise, horseradish, and Creole seasoning, stirring until smooth. Cover and refrigerate until ready to use.

In a large skillet, heat 1 tablespoon oil over medium heat. Add yellow onion, bell pepper, celery, garlic, and salt; cook, stirring occasionally, until lightly caramelized and softened, 10 to 15 minutes. Transfer to a large bowl, and let cool slightly. Wipe skillet clean.

Add green onion, crackers, and remaining ¼ cup mayonnaise and eggs to onion mixture, stirring until well combined.

In the work bowl of a food processor, pulse crawfish until coarsely chopped. Add to onion mixture, stirring until well combined. Form into 8 (3-ounce) ½-inch-thick patties.

In same skillet, heat 1 tablespoon oil over medium-high heat. Add half of patties; cook until golden brown on both sides, 5 to 7 minutes. Remove from skillet, and wipe skillet clean. Repeat with remaining 1 tablespoon oil and remaining patties.

Spread horseradish sauce on 1 side of all bread. Place 1 slice of each cheese on top of horseradish sauce. Place 2 crawfish patties on half of slices of bread, and top with remaining slices, cheese side down.

In same skillet, melt half of butter over medium heat. Add 2 sandwiches, and cook until golden brown on both sides, heated through, and cheese is melted, 5 to 7 minutes. Remove from skillet, and repeat with remaining butter and remaining 2 sandwiches. Serve with chips and pickle spears.