Preheat the oven to 375°F and set a rack in the middle position.
Line a baking sheet with heavy-duty aluminum foil. Grease foil with 1 tablespoon of the olive oil.
Clean the mushrooms with a damp towel, remove stems, and finely chop the stems.
Place mushroom caps on the prepared baking sheet, open side up, drizzle with 1 tablespoon olive oil, ¼ teaspoon salt, and a few grinds of pepper, and set aside.
In a medium skillet over medium heat, warm the remaining 1 tablespoon oil.
Add the sausage and cook, breaking it up into small pieces with a wooden spoon, until browned, about 5 minutes.
Add the chopped mushroom stems, shallot, and garlic; cook until softened, 3 to 4 minutes more.
Add the cream cheese and mix until evenly combined.
Remove from the heat and stir in breadcrumbs, Pecorino (or Parmigiano Reggiano), parsley, a scant ½ teaspoon salt, and ¼ teaspoon pepper until well combined.
Spoon the sausage mixture into each mushroom, pressing it gently so it fills the cavity and then mounding extra on top.
Arrange the stuffed mushrooms on the prepared baking sheet and bake for 25 to 30 minutes, until the mushrooms are tender and the tops are golden.
Sprinkle with additional parsley and serve warm.