Thirty to sixty minutes before ready to bake the pizzas, place a baking stone on a rack in the lowest level of the oven; turn on oven to maximum temperature, 500°.
Wipe mushrooms clean with damp paper towel; discard stems of shiitake mushrooms; cut mushrooms into thin slices.
In a big pan, heat the oil with the garlic over medium heat; add in the mushrooms, thyme, salt, and pepper to taste; cook, stirring often.
When the mushrooms liquid has evaporated and the mushrooms are browned, add the parsley; let cool.
Place pizza dough on a floured surface; Using your fingers, stretch and flatten dough out into a 12-inch circle.
Dust a pizza peel with flour; place dough on the peel, shaking it once or twice to make sure it does not stick—if it does, lift the dough and dust the peel with more flour.
Spread the cheese slices over the dough leaving a ½ inch border all around; top with the mushrooms.
Slide pizza onto prepared baking stone; bake pizza 6-7 minutes or until the edges are puffed and the crust is crisp and golden brown.
Remove pizza to a cutting board; cut into wedges and serve immediately.