Remove the stems from the mushrooms. Finely mince the mushroom stems and set aside. Add the mushroom caps to the prepared baking sheet, drizzle with 2 tablespoons of olive oil, and toss to coat. Set aside.
In a large skillet over medium heat, add 1 tablespoon of olive oil. Once hot, add the Italian sausage, breaking it up into small pieces and cooking until browned and crisp, about 5-7 minutes. Once the sausage is fully cooked though, use a slotted spoon, and transfer the sausage to a medium mixing bowl.
To the same skillet, add the remaining tablespoon of olive oil, shallot, and chopped mushroom stems. Stir the mixture to coat the mushroom stems and shallot in the oils, and spread into an even layer. Allow the mushroom stems to sear and get golden brown, about 3 minutes before stirring.
Continue to cook until most of the liquid from the mushrooms has cooked out and the shallots are translucent, about 1 more minute. Add in the garlic, white part of the scallions, dried sage, salt, and pepper. Sauté until the garlic is fragrant and the spices smell toasted, about 1 minute. Pour in the sherry vinegar, deglazing the pan and stir to combine. Transfer the contents of the skillet into the medium bowl with the sausage.
To the bowl, add the green parts of the scallions, softened cream cheese, parmesan, parsley, fresh thyme leaves, lemon zest, panko breadcrumbs. Stir until the filling is combined. Set aside.