Wipe off mushrooms with a damp paper towel. Twist stems of mushrooms to remove (or use a paring knife).
Chop mushroom stems finely and set aside.
Cook and crumble sausage finely. Remove to a plate to cool.
Reduce heat to MED-LOW and add onions and garlic to the same skillet with sausage grease. Saute for 3 minutes.
Pour in wine to deglaze pan, using a wooden spoon to scrape the pan to release the brown sausage bits, and allow most of the liquid to cook off.
Stir in chopped mushroom stems and thyme and saute for 2 minutes. Add salt and pepper to taste. Set mixture aside on a plate to cool.
In a bowl, combine softened cream cheese, egg yolk, and Parmesan cheese. Stir to combine well.
Add cooled, cooked sausage and cooled onion mixture. Stir well, cover with plastic wrap and refrigerate for 30 minutes or so to firm up.
Preheat oven to 350 degrees, line a baking sheet with parchment paper.
Using a spoon or teaspoon measuring spoon, add mixture to the top of the mushrooms, pressing slightly to fill the cavity. Add a little extra to create a mound on top of the mushroom.
Bake for 25 minutes, or until golden brown. Broil the last minute if you want a more golden brown topping.
Let cool a few minutes, sprinkle with chopped fresh parsley, and serve.