Heat oil in medium skillet over medium heat. Add mushrooms and sauté until golden brown, 10-15 minutes. Season to taste with salt and pepper. Transfer to a plate or bowl. They can be refrigerated until needed.
Turn skillet to medium low heat and add 1 tablespoon olive oil. Sauté the onions until soft and golden brown. If the onions become dry and start to brown to quickly, add 1-2 tablespoon water to the pan. Transfer to a bowl or plate and refrigerate if needed until you're ready to use them.
Place a pizza stone on the middle rack of the oven and preheat to 500°F for one hour.
Cut the dough into four pieces. Work with one piece at a time and the the others covered. Roll and/or stretch into a circle. Lay on a wooden cutting board, pizza peel or rimless baking sheet that has been covered with cornmeal or semolina. Lightly brush with olive oil and cover with the onions, mushrooms and crumbled goat cheese. Add freshly ground pepper if desired.
Transfer pizza to the pizza stone in the oven. Bake until crust and cheese are starting to brown, 10-15 minutes.
Remove from oven and cut into wedges.