Heat 2 tablespoons olive oil in a large skillet over medium heat. Add shallots and 1/4 teaspoon salt. Sauté, stirring frequently, until shallots caramelize, 10 to 12 minutes. Set aside.
In another skillet, heat 2 tablespoons olive oil over medium-high heat. Add mushrooms, thyme and 1/4 teaspoon salt. Cook until mushrooms wilt and liquid evaporates, 5 to 7 minutes.
Divide dough and form two pizzas. Brush each with 1 tablespoon olive oil. Strew mushrooms and crumbled goat cheese evenly over pizzas.
Slide one pizza onto the hot stone. Bake until crust is golden, 8 to 10 minutes; repeat with the second pizza. If using a heavy baking sheet, increase cooking time to 10 to 12 minutes.
Strew caramelized shallots over tops of pizzas; season with pepper. Cut into wedges.