For the Cajun-spice patties: Prepare a grill or grill pan, and heat to medium-high heat.
Gently mix the ground beef, Cajun seasoning, and scallions, and then divide the mixture into four 1/2-inch-thick patties.
For the remoulade: Combine the mayonnaise, mustard, hot sauce, parsley, and celery in a small bowl. Refrigerate until needed.
Grill the patties, flipping once, to desired doneness, about 4 minutes per side for medium. Brush the onion and green pepper with the oil, and grill the vegetables until lightly charred and tender, about 5 minutes total.
Toast the buns on the grill for a minute or so. Spread both the top and bottom buns with the remoulade. Put a patty on the bottom bun, add some onion and bell pepper, then add the top bun.