For the mushrooms: If the mushroom caps are very rounded, trim a very thin slice from the top of the caps to help them sit upright for baking. Toss the mushroom caps with the oil and 1/4 teaspoon salt in a large bowl.
For the filling: Heat the butter in a large skillet over medium-high heat. Add the chopped mushroom stems, celery and onion and cook, stirring occasionally, until just softened, 6 to 8 minutes. Scrape into a large bowl, season with 3/4 teaspoon salt and set aside to cool.
Add the mozzarella, sausage, cranberries, sage and thyme to the cooled mushroom stem mixture. Stir in the cornbread until well moistened.
Pack each mushroom with an equal amount of filling and transfer the filled mushrooms to a rimmed baking sheet. Bake until the mushrooms are tender and the filling is brown and crispy on top, about 15 minutes. Let cool 5 minutes before removing from the baking sheet.
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