For the red wine-rosemary butter: Place the wine and the rosemary sprig in a small saucepan over medium heat. Bring to a boil, then lower the heat and simmer until the wine has reduced to a syrup and measures approximately 2 tablespoons, about 20 minutes. Let cool to room temperature before removing and discarding the rosemary sprig. Place the butter, the chopped rosemary and the salt in a food processor and pulse to combine. Add the wine syrup and process until fully integrated, about 1 minute. Set aside.
For the strip steak: Preheat a grill or grill pan to high heat. In a small bowl, combine the olive oil, rosemary, salt and garlic, and rub the mixture all over the steaks. Place the steaks on the grill and sear for 4 minutes without moving to form a golden brown crust. Flip the steaks and sear on the reverse side until crusty on the outside and medium-rare at the center, another 3 minutes. Remove the steaks to a plate and let rest for 10 minutes.
Meanwhile, drizzle about 1 teaspoon olive oil over the lemon halves and place them, cut side down, on the grill. Grill until charred and fragrant, 3 minutes.
To serve: Transfer each steak to a plate and top with a dollop of red wine butter and a squeeze of the charred lemon.