Combine Italian dressing, garlic powder, oregano, and basil into a plastic bag. Add the chicken breasts and mix well. Marinate in refrigerator overnight.
Preheat grill over medium heat.
Make foil pouches of sliced red onion with butter and another pouch of sliced mushrooms with butter. Place onto hot grill and cook until tender. Drizzle portobello caps with olive oil and place on hot grill to soften. Remove chicken from marinade and discard marinade. Grill chicken breasts until cooked throughout, sprinkling with Cajun seasoning and garlic powder, set to side of grill to keep warm. At same time, grill 4 strips of bacon.
Spread the cooked portobellos with a thick layer of cream cheese. Place 1 cooked chicken breast onto each, spoon on some of the sliced mushrooms, and finally some of the sliced onions. Place a strip of bacon onto that, and then cover with sliced mozzarella and shredded Monterey Jack cheese, put back on grill to melt cheese, and serve with a side of dirty rice.