1. In a large skillet, heat 1 tbsp oil over medium-high heat. Add onion and cook, stirring frequently, 2 minutes. Crumble sausage into skillet and cook, breaking up with a fork, until browned, about 5 minutes. Stir in panko and basil; cook 1 minute more. Let cool, then finely chop.
2. Rinse pork and pat dry. Cut a 3-inch-deep horizontal slit, leaving 1/2 inch uncut at each end, down center of pork, creating a pocket. Stuff sausage mixture into pocket. Tightly tie pork with kitchen string to enclose stuffing. Transfer to an airtight container; generously season with salt and pepper. In a small bowl, combine remaining 3 tbsp oil, the rosemary and garlic; spread over pork, seal container and refrigerate at least 4 hours or overnight.
3. Let pork stand at room temperature while preparing grill. Heat grill to medium (or heat oven to 500 degrees F). Grill pork, turning every 10 minutes, until an instant-read thermometer inserted in center by 2 inches registers 145 degrees F*, 25 to 30 minutes (or roast in oven 50 minutes to 1 hour). Transfer to a cutting board and let stand 5 minutes before carving into 3/4-inch thick slices. Makes 8 to 10 servings.