Mix all rémoulade ingredients until well combined. Set aside.

Preheat the grill or broiler. Mix turkey, Cajun seasoning, minced garlic, and teriyaki sauce until well combined, and form into six round patties. Sprinkle the tops of the patties with kosher salt and pepper.

Grill or broil patties for six minutes per side until cooked through completely and slightly brown on the outside. Allow to rest for five minutes.

Place lettuce and tomato on each bun, and top with a burger. Spoon rémoulade sauce generously over each burger. Add sliced onions and alfalfa sprouts. Serve warm.