Combine the mushrooms and shallot in a bowl and toss with 1 tablespoon olive oil and salt and pepper to taste. Spread on a baking sheet and place under the grill until the mushrooms are starting to char, about 10 minutes.
Meanwhile, in another bowl, mix together the balsamic, lemon juice and 2 teaspoons olive oil. Preheat oven to 180 degrees C.
Scatter the mozzarella over the pizza base and distribute the mushrooms evenly. Drizzle with the vinaigrette. Bake in the preheated oven 20 minutes or until the base is cooked. Scatter over the parsley before serving.