Remove the ribeye steaks from the refrigerator about 20 minutes before cooking so they warm slightly. Pat the steaks completely dry with paper towels to remove surface moisture, then season both sides evenly with kosher salt and black pepper, pressing gently so the seasoning adheres.
Preheat the oven to 400°F (205°C) and set a rack in the center position so heat circulates evenly around the skillet. Preheating fully helps the steaks finish at a steady temperature once transferred from the stovetop.
Place an oven-safe skillet over medium-high heat and heat for several minutes until very hot. Add the neutral oil and swirl to coat the bottom of the pan. The oil should shimmer before you add the steaks.
Carefully place the steaks into the hot skillet and let them sear without moving for 3 to 4 minutes. This undisturbed sear builds a deep brown crust on the first side.
Flip the steaks using tongs. Add the butter and smashed garlic cloves to the skillet. As the butter melts, tilt the pan slightly and spoon the melted butter over the steaks to baste for 2 to 3 minutes, helping flavor the surface and control browning.
Using oven mitts, carefully transfer the skillet to the preheated oven. Finish cooking for 6 to 10 minutes depending on thickness and desired doneness. The oven heat cooks the interior evenly while the crust stays intact.
Check the internal temperature with an instant-read thermometer inserted into the thickest part of the steak. Remove at 125°F for rare, 130 to 135°F for medium-rare, or 140°F for medium. Remove a few degrees early because the temperature will rise slightly during resting.
Transfer the steaks to a plate or cutting board and rest for at least 5 minutes. Resting redistributes juices throughout the meat for a more tender, juicy steak.
Slice the ribeye steaks against the grain or serve whole. Spoon a small amount of the pan butter over the top if desired, then serve immediately while warm.