Preheat the oven to 350 degrees F. Place the first six ingredients in a bowl and mix well to combine. Stuff and mound the crab mixture inside the mushroom caps. Set the mushroom caps in a parchment-lined or lightly-oiled baking dish; sprinkle with the parmesan cheese.
Preheat the grill to medium-high. Bake the mushrooms 15 to 20 minutes, or until tender and lightly brown on top. Season the steaks with salt and pepper. Grill the steaks, lightly oiling the surface if not non-stick, 2 to 3 minutes per side for rare; 3 to 4 for medium rare; and 5 minutes for medium. Set the steaks and mushrooms on plates and serve.