Preheat oven to 350 degrees.
Remove and discard any stems and tough leaves from the greens. Chop the leaves in a food processor. If greens are large, blanch briefly and drain well.
Remove and chop the stems of the mushrooms and set aside.
Melt 1 ⁄ 2 cup butter in a Dutch oven or deep skillet over medium heat, and dip the mushroom caps into it until well coated on all sides. Place them top side down on a greased rimmed baking sheet.
Heat remaining butter in the skillet; add the onions, garlic, and chopped mushroom stems, and sauté until very soft, about 10 minutes.
Add the greens and cook a few minutes until wilted; drain off and reserve some of the liquid.
Add the breadcrumbs and mustard, and mix well. Add back some of the reserved liquid if the mixture is so dry that it doesn’t stick together easily.
Season to taste with salt and pepper.
This filling can be made several days in advance or frozen and defrosted before stuffing mushrooms.
Fill each buttered mushroom cap (up to 1 day in advance) with this mixture, mounding it high, and sprinkle with Parmesan cheese. Bake 10 to 15 minutes.