Place prepared pizza dough in a 16x12x1-inch pan or a round pizza sheet. Arrange cheese slices on top of dough.
In large skillet, cook onions, covered, in 2 tablespoons olive oil over medium low heat for 13-15 minutes, or until onions are tender, stirring several times. Uncover. Now cook and stir over medium high heat for 5-10 minutes longer or until onions are golden. Remove from pan and set aside.
In the same skillet, combine mushrooms, remaining 1 tablespoon olive oil, garlic, and rosemary. Cook over medium heat until mushrooms are tender and begin to caramelize; drain well. Spoon mushroom mixture over cheese. Top with the onions.
Bake in the preheated oven for 25-30 minutes or until bottom of crust is slightly crisp and brown. Cool in pan on a wire rack for 5 minutes. Season with salt and pepper to taste. Sprinkle with parsley. Cut the pizza into 4" squares and serve immediately. Enjoy!