Remove steak from the refrigerator 30 minutes before cooking to sit out at room temperature.
Heat your grill for approximately 10 minutes before cooking. Ensure you have a rack that provides indirect heat or that one side of the grill is not turned on. Once the grill is heated, lightly oil the grill grates.
Season the steak with the kosher salt and pepper and place on the grill.
Let the steak cook for 3-4 minutes and then turn 90 degrees to allow for grill marks. Cook an additional 3-4 minutes. Flip to cook other side and repeat. Once marked on all sides, move the steak to the indirect heat rack. Let cook for 7-9 minutes and flip halfway through cooking.
While steak is cooking, toss eggplant halves in oil and season. Place on grill, cook until soft, flipping halfway through cooking, about 4-5 minutes each side. At the same time, lightly oil the green onions and place on grill. Cook until nicely charred, about 3-4 minutes. Once the eggplant is finished cooking, chop into 1-inch pieces. Cut the green onions into 1-inch pieces and toss the eggplant, green onions, olive oil, balsamic vinegar and thyme together and set aside in a serving bowl. Season to taste if needed.
Cook the steak to your desired temperature. Color is not an accurate indicator of final cook temperature. Ensure final cook temperature reaches 145 F for 3 minutes. Let steak rest for 5 minutes.
Cut the steak off the bone and slice the steaks. Place the sliced steak around the bone and serve with the charred eggplant and green onion.