Make the marinade by pouring the chicken broth, minced garlic, lemon juice, Creole mustard and hot sauce into a rectangular plastic container with a fitted lid. Stir until well-combined.
Submerge the chicken in the marinade. Marinate for at least 3 hours, but preferably overnight.
Make a double portion of the Remoulade sauce. Reserve half of the sauce in a covered dish for serving alongside the finished Cajun grilled chicken.
Preheat your gas grill or prepare your charcoals for grilling over medium heat, at about 350 degrees F.
Carefully remove the chicken pieces from the marinade, and shake gently to release any excess marinade. Discard the marinade.
Grill the chicken pieces for about 40 minutes. Then, baste them about 5 minutes or so until they are ready, when a digital thermometer inserted into each piece reads 165 degrees F.
Alternately, bake the chicken for 40 minutes at 375 degrees F. Then, baste for the remaining 20 to 30 minutes of roasting time.