Heat half the oil in large skillet set over medium-high. Add the mushrooms and garlic. Cook the mushrooms until tender, about 5 minutes, and then remove from the heat.
Preheat your barbecue to medium-high. Brush the pita bread lightly with the remaining olive oil. Grill the pita until nicely coloured on one side, remove from the heat and set on work surface. Turn the barbecue to low. Spread the grilled side of each pita with tomato sauce and then top with the mushrooms, cheese, bell pepper and black pepper. Carefully set the pita back on the grill, cheese side up. Close the lid and cook 3 to 4 minutes, or until lightly charred on the bottom and the toppings are heated through. Plate the pita, sprinkle with basil and serve.
Note: When preparing the shiitake mushrooms, discard the tough stem and only slice the cap. When preparing the oysters mushrooms, trim off the tough bits of lower stem. Greek-style pita is available in the deli department.