Preheat a lightly oiled grill to medium. In a large skillet, heat oil to medium heat. Add mushrooms and cook, stirring occasionally, until water evaporates and mushrooms are golden brown, about 12 min. Stir in thyme and pepper, and set aside.
Lay crusts down on lightly oiled grill. Cook for 2 min., or until bubbly and rigid. Using tongs, turn crusts over. Divide the herb paste, mushrooms, and cheese between the two crusts. Turn heat down to low, close lid and cook until cheese is melted, about 2 min.
Remove from grill and place prosciutto over top of each pizza. Cut each pizza into six slices and serve.
To make pizzas indoors, heat pizza stone in the oven at its hottest setting. Dress rolled dough with herb paste, mushrooms and cheese. Slide onto stone using a pizza peel or floured rimless tray. Cook for approx 5 min., or until cheese is bubbling and crust is lightly browned. Top with prosciutto and serve.