Use your hands to gently pull off and remove the mushroom stems. Set aside, do not discard. Drizzle olive oil on a large rimmed baking sheet. Finely chop the mushroom stems.
Heat a large skillet over medium-high heat. Melt butter, then add Italian sausage. Use a spatula to break the sausage up as it cooks until browned on all sides. Add the chopped mushroom stems, shallot, garlic, Kosher salt, and black pepper. Cook.
To the sausage mixture, add cream cheese, shredded asiago (white cheddar or Parmesan is fine) – reserve the remaining cheese – and chopped fresh sage. Stir until very well mixed.
Fill each mushroom cap with a generous cream cheese mixture so that it is a heaping mound on top. Sprinkle each with the remaining shredded cheese.
Cool slightly, then serve warm with chopped parsley and a pinch of flaky sea salt on top.