Preheat your oven to 350° F.
Toss the mushrooms with 3 tablespoons of olive oil, balsamic vinegar, salt, and pepper. Spread out on a sheet pan and bake for 30 minutes. Set aside and cool.
Crumble the sausage in medium pan and sauté over medium high heat until golden brown. While the sausage cooks, break it up into small pieces. When it’s completely cooked through, remove from heat and set aside to cool.
Add the remaining 1 tablespoon of olive oil to the sauté pan and add the onions. Cook until golden and nicely caramelized, about 15 to 20 minutes. Then add in the garlic and cook for a minute.
Now add the cooked sausage, onions and garlic, cream cheese, salt and pepper, and 3 ounces of shredded Asiago cheese to a bowl. Mix well with your hands. If there are any large pieces of sausage, break them into small pieces with your hands.
Heat your oven to 375° F. Line up your mushrooms in a greased baking dish with the lids down, core up. Stuff each mushroom with a generous portion of the creamy sausage mixture. Sprinkle the mushrooms with the remaining Asiago cheese. You can make them ahead of time up to this point. Store in the refrigerator until ready to finish cooking.
Bake them for about 45 minutes or until the cheese on top is golden brown.