Prepare the grill or barbecue (medium-high heat). Sprinkle the steaks with salt and pepper and drizzle with the 3 tablespoons of olive oil. Grill to desired doneness, about 5 minutes per side for medium-rare. Transfer the steaks to a cutting board. Tent with foil and let stand 10 minutes.
Meanwhile, melt 2 tablespoons of butter in a heavy large saucepan over medium-high heat. Add the onions and saute until tender, about 5 minutes. Season with salt. Add the garlic and oregano and saute until fragrant, about 30 seconds. Stir in the tomato paste and cook for 2 minutes, stirring constantly. Whisk in the wine. Simmer until the sauce reduces by half, stirring occasionally, about 10 minutes. Remove the skillet from the heat. Strain the sauce into a small bowl, pressing on the solids to extract as much liquid as possible. Discard the solids in the strainer and return the sauce to the saucepan and bring back to a slow simmer. Cut the remaining 4 tablespoons of butter into small 1/2-inch chunks and whisk in the sauce a little at a time. Season the sauce, to taste, with salt and pepper.
Thinly slice the steaks across the grain. Divide the steak slices among 6 plates. Drizzle the sauce over the steak, drizzle a little more extra-virgin olive oil and serve.