Preheat the oven to 350 F. Line a rimmed baking sheet with parchment paper or foil.
Trim the very ends from the mushroom stems, then remove the stems and finely chop them. Place the mushrooms hollow side up in a baking dish or on a rimmed baking sheet.
Heat the olive oil in a large skillet over medium heat. Add the onion and chopped mushroom stems and sauté for 5 minutes until softened and golden. Add the garlic and sauté for another 2 minutes, until golden. Add the white wine and cook for another 3 minutes until the wine has mostly evaporated. Add the spinach and thyme and cook until wilted, about 1 minute, then season with salt and pepper. Add the cream, provolone, and 1/2 of the Parmesan. Stir until everything is well blended and thick and the cheese is pretty well melted, about 3 minutes. Remove from the heat and stir in the panko.
For smaller mushrooms, scoop a rounded teaspoon of the mixture into the hollow of each mushroom cap and mound it attractively. For larger mushrooms, use a tablespoon of the filling. Sprinkle over the remaining Parmesan evenly over the top. Bake smaller mushrooms for about 15 to 20 minutes until hot throughout and the tops are golden brown. Bake larger mushrooms for about 30 minutes.
Serve hot, sprinkled with fresh parsley.
Notes
You can freeze uncooked stuffed mushrooms for up to 3 months. Fill the mushrooms and then place them on a plate or a baking sheet. Place them in the freezer and freeze for 1 hour until very firm. Gently transfer them upright to a freezer-proof container, tuck plastic wrap all around them, then cover and freeze. You should not defrost them before baking. Thawing the mushrooms before baking will result in soggy stuffed mushrooms, so bake them straight from the freezer.