Gruyère Mushroom Pizza With Balsamic Glaze

Name: Gruyère Mushroom Pizza With Balsamic Glaze
Ingredients: 4 tablespoons extra-virgin olive oil , divided; 2 ( 8 - ounce ) balls pizza dough , store-bought or homemade; 12 ounces assorted mushrooms , cleaned and thinly sliced or torn into bite-sized pieces; Kosher salt; Freshly ground black pepper; 1/2 cup loosely packed flat-leaf parsley , finely chopped, divided; 1 cup heavy cream; 1 clove garlic , peeled; 8 ounces low-moisture mozzarella cheese , shredded; 4 ounces Gruyère cheese, shredded; 2 tablespoons balsamic glaze , for drizzling, plus more if needed