In a mixing bowl, mix together 1 can of beer, 1½ cups BBQ sauce, and ½ cup of Cajun seasoning.
Follow the manufacturers instructions on filling your marinade injector. Inject chickens in several places at least 1 inch deep.
Place chickens in zip lock bags, divide any remaining marinade between the bags and pour in. Seal bags and place in the fridge to marinate for 1-3 hours, or even overnight.
While the chickens are marinating, prepare the Cajun BBQ Sauce. In a saucepan, add 1 cup BBQ sauce, ½ cup beer, butter, and 1 tablespoon Cajun seasoning. Place on medium low heat and mix together, simmer for 5 minutes. Remove from heat, allow to cool, then cover and place in the fridge until serving. Use the beer from the next step.
Open the remaining 3 cans of beer, pour out about ½ cup beer from each can, then use a can opener and punch several holes in the tops of the cans. Spoon about 1 tablespoon Cajun seasoning into each can and insert 4 sprigs of thyme into each can.
Prepare your grill for indirect cooking on medium high heat (180° C / 350° F).
Remove chicken from bags and discard bags and marinade. Place 1 can into the cavity of a chicken and place the can upright, arrange the legs to hold the chicken upright. Repeat with other chickens.
Place chickens upright on the grill over indirect heat. Cover grill and allow the chickens to cook for about 1 ½ hours or until a meat thermometer reads 83° C (180° F) internal temp. Cover chickens with foil if they become too browned during the cooking time.
Remove chickens from the grill and allow to rest 10 minutes. Use tongs to carefully remove the cans.
While the chickens are resting, reheat the Cajun BBQ Sauce.
Cut each chicken into quarters, serve with the hot Cajun BBQ Sauce and sides of your choice, enjoy.