Combine the cheeses (except for the Parmesan cheese), garlic, herbs, white wine, ½ tablespoon olive oil, and ½ teaspoon kosher salt in a medium bowl and mix together.
Toss the mushroom caps with 1 tablespoon olive oil and ½ teaspoon kosher salt to evenly coat.
Spoon each mushroom cap full of the cheese mixture, heaping it slightly over the top of the mushroom, then top with a bit of the Parmesan cheese.
