Preheat oven to 400° F with a rack in the center position. Prepare a rimmed baking sheet, either by setting an oven-safe wire rack on top (you can place a sheet of parchment paper under the rack for easy clean up) or by simply lining the pan with a sheet of parchment paper. If baking without a wire rack, the mushrooms will be sitting in any liquid they release as they bake so the stuffed mushrooms may be slightly less firm.
Pull out the mushroom stems and chop them finely to use in the filling. Set mushroom caps aside.
In a medium skillet, heat the olive oil over medium heat. Add the chopped mushroom stems and onion and sauté, stirring often, for about 5 minutes, until softened. Stir in the garlic and cook, stirring, for 30 seconds. Remove pan from the heat. Let cool for 10-15 minutes.
In a medium mixing bowl, stir together the cream cheese, ¼ cup grated Parmesan, fresh thyme, salt, pepper and red pepper flakes. Stir in the sautéed mushroom stems, onions and garlic from the skillet.
Spoon the mixture into the mushrooms and set them, filling side up, on the prepared baking sheet.
In a small bowl, stir together the Panko bread crumbs and 1 tablespoon Parmesan. Stir in the melted butter. Sprinkle the bread crumb mixture on top of each stuffed mushroom.
(When transferring the pan to the oven, do so carefully or the mushrooms may roll over.) Bake for 15-20 minutes, until mushrooms are tender and topping is golden brown. Serve warm.