Pat the ribeye steaks dry with paper towels. Season all sides evenly with kosher salt and black pepper; if time allows, refrigerate uncovered 40–60 minutes for a quick dry brine.
Adjust the oven rack to the middle position and heat the oven to 450°F (230°C). Set a 12-inch cast-iron skillet over medium-high heat until very hot, 4–5 minutes.
Add the oil and heat until shimmering and just beginning to smoke, about 30 seconds.
Lay the ribeye steaks in the skillet and sear without moving until a deep brown crust forms, 2–3 minutes.
Flip and sear the second side for 2 minutes.
Add the butter, garlic, and thyme to the skillet. Tilt the pan and baste the steaks continuously with the foaming butter for 60–90 seconds.
For steaks 1.25–1.5 inches thick, transfer the skillet to the oven and roast 3–6 minutes, until an instant-read thermometer inserted sideways into the center registers 120–125°F for rare, 130°F for medium-rare, or 135°F for medium. For thinner steaks or if already near temperature, continue basting on the stovetop over medium heat to the same internal temperatures, 1–3 minutes.
Transfer the steaks to a wire rack or warm plate and rest 5–10 minutes; the temperature will rise about 5°F and juices will thicken.
Slice across the grain or serve whole. Sprinkle flaky sea salt before serving and spoon any pan juices over.