Preheat your barbecue or indoor grill to medium-high. Brush the steaks very lightly with 1 Tbsp. oil and season with salt and pepper. Place the remaining oil and butter in a large skillet and set over medium heat. When the butter is melted, add the mushrooms, garlic, and rosemary and cook until the mushrooms are tender, about 5 to 6 minutes. While the mushrooms are cooking, grill the steaks to the desired doneness, allowing approximately 2 to 3 minutes per side for rare to medium-rare, and 3 to 4 minutes per side for medium. When the mushrooms are tender, add the lemon juice and season with salt and pepper. Divide the steaks and mushrooms among plates and serve.
Note: The stems from shiitake mushrooms need to be removed and discarded before the mushroom is sliced. The tough lower portion of the stem on oyster and portobello mushrooms should be trimmed and discarded.