Remove stems from mushrooms; set caps aside. Coarsely chop the stems.
Bake mushroom caps for 15 minutes at 450F, flipping them halfway through. Once done, turn the oven down to 350F.TIP - Mushrooms release a lot of liquid when they cook. Since our filling does not have breadcrumbs there's nothing to soak up that liquid. They need to be roasted first.
Cook chopped mushroom stems and onions in 1 Tbsp of oil in a large skillet over medium-low heat for 5 minutes or until tender.
Stir in parsley, basil and Vermouth. Simmer 10 minutes or until wine is absorbed; remove from heat; cool for 5 minutes. I spread mine out on a plate for fasting cooling.
Stir in cheeses. Fill mushroom caps with cheese mixture and place in an 8×8-inch baking pan. Bake the stuffed mushrooms at 350F for 15 minutes or until filling is hot and cheese is melted.