Whisk all the marinade ingredients together in a large bowl. Set aside. Transfer half of the marinade into 1 shallow non-reactive pan or bowl and reserve the other half of the marinade for the mushrooms.
Trim excess fat from the steaks and make small cuts with a knife into the fat around the edge of the steaks. Marinate steaks for at least 30 minutes at room temperature, or up to 6 hours in the refrigerator.
Preheat a grill to high heat.
Place steaks on the hot grill for 3 minutes on each side to sear and create nice grill marks. Reduce the grill temperature to low and close the lid. Continue cooking steaks an additional 5 to 10 minutes for medium-rare, turning them halfway through. Let steaks rest for 5 minutes before serving.
Meanwhile, dip the mushrooms in the reserved marinade and place on the grill, curved side down. Put the leftover marinade from the steaks and mushrooms in a small saucepan and bring to a boil for at least 3 minutes and cook until reduced slightly. Lightly coat the serving platter with extra-virgin olive oil and whole leaves of fresh rosemary, oregano, and basil. Transfer the steaks and mushrooms to the platter and drizzle with the sauce.