Remove the steaks from the refrigerator 20 minutes before cooking to bring them to room temperature; pat them completely dry with paper towels.
Season both sides of the steaks generously with the sea salt and black pepper.
Trim the woody ends off the asparagus and toss the spears with the olive oil and a small pinch of salt.
Heat a large cast-iron skillet over medium-high heat until it is extremely hot and just beginning to smoke.
Place the asparagus in the skillet and sear for 4-5 minutes, turning occasionally, until they are bright green and slightly charred; remove and set aside.
Place the steaks in the same hot skillet and sear for 3-4 minutes without moving them to develop a deep brown crust.
Flip the steaks and immediately add the butter, rosemary sprigs, and smashed garlic cloves to the pan.
Tilt the pan slightly so the melting butter pools at the bottom with the aromatics, then use a large spoon to continuously baste the steaks with the foaming butter for 2 minutes.
Check for your desired level of doneness (130-135°F for medium-rare) and transfer the steaks to a cutting board.
Let the meat rest for 5-8 minutes to allow the juices to redistribute before slicing and serving alongside the charred asparagus.