Method:
Preheat grill. Brush both sides of mushrooms with oil and season with salt and pepper, then arrange on grill, stem side down, and tent with foil. Cook for 3 to 4 minutes, then flip and grill on the other side, again covered with foil, until almost tender, 2 to 3 minutes more. Transfer mushrooms to a large platter, stem side up.
Fill mushrooms with pesto, tomatoes, basil and turkey then top with cheese and pepper flakes. Return to grill, tent with foil and cook, filled side up, just until cheese is melted and toppings are warmed through, 2 to 3 minutes more. Transfer to plates and serve.