Prepare a stove-top griddle or outdoor grill.
Combine the tomatoes, thyme, 2 teaspoons of olive oil, the roasted garlic, and salt and pepper in a small bowl and toss. Set aside.
Brush the top and bottom of each mushroom with the remaining oil. Sprinkle with salt and pepper and grill until the mushrooms are tender, about 4 minutes per side.
Place the mushrooms stem side up on a cutting board. Arrange an equal amount of the tomato mixture on 4 mushrooms. Top each with a shaving of Parmigiano-Reggiano.
Arrange a slice of prosciutto on each of the remaining mushrooms and spread 1 tablespoon of the pesto on top of each prosciutto slice.
Arrange 1 of each pizza on each of 4 dinner plates and serve.