Heat barbecue grill until very hot. Prepare garlic for roasting be removing outer, papery layer. Slice off top of garlic so that cloves are exposed. Drizzle with 1/2 teaspoon of olive oil, wrap tightly in double thick square of foil and place directly on rack of hot grill. Repeat this same procedure with onions (warp individually).
Rub portobello caps, stems, criminis with olive oil, salt and pepper, add to grill with onions and garlic. Remove portobello caps, stems, and criminis from grille after lightly browned (do not overcook, approximately 3 to 5 minutes on each side). Cut portobello stems and criminis into 1/4-inch strips, set aside, keep warm while cooking chicken.
On the stove (or in a pan on the barbecue grill), heat a saute pan until hot, add 1 tablespoon of olive oil, add the roasted garlic and saute for 15 seconds (do not brown). Reduce heat to medium high, toss chicken strips in flour, and add to garlic mixture. Season with salt and pepper, add red pepper flakes and saute quickly until lightly golden browned and chicken begins to stick to pan. Add wine, butter and reduce heat to medium. Continue cooking until light sauce develops (approximately 5 minutes), toss in herbs and continue cooking for 1 to 2 minutes then remove from heat.
To assemble, place mushroom caps on plates, sprinkle with shredded Parmesan cheese. Gently toss together chicken strips, pepper strips, onion strips, and mushroom strips and pile on portobello cap. Add 2 to 3 Parmesan curls, decorate with a basil sprig and serve warm.