Preheat the oven to 400 degrees. Spread the pizza dough out to cover a large, parchment-lined baking sheet, then brush with 2 tablespoons of the olive oil and sprinkle with the garlic. Bake for 8 minutes.
Meanwhile, thinly slice the shiitake and white mushrooms. In a large nonstick skillet, heat the remaining tablespoon of olive oil over high heat and saute the mushrooms until golden, 8 to 10 minutes. Season to taste with salt.
Sprinkle the baked dough with the Swiss cheese, then the mushrooms and the mozzarella. Sprinkle a little more salt over the topping and return the pizza to the oven. Bake until the cheese is golden and bubbly, about 15 minutes. Cut into squares and serve.