Pat the steaks dry and season both sides with salt and pepper. Heat olive oil in a cast iron pan over medium-high heat. Once it’s smoking, add the steaks and set a timer for 4 minutes. Don’t move or flip the steaks during this time!
After 4 minutes, flip and add the butter and thyme to the pan. Set a timer for 4 minutes. While the steak is cooking on this side, tip the pan and use a spoon to scoop the juices in the pan over the steaks. Do this the entire time the steak is cooking, basting it with the butter/oil/thyme mixture. When the timer goes off, use a meat thermometer to check the internal temperature (these cook times and based on our preference for rare steaks, btw. Adjust time accordingly to your tastes!).
Remove the steaks and put them on plates to rest for 5 minutes. While they are resting, add the broccoli to the pan and squeeze the lemon juice over it. Saute and allow to crisp and char until it turns bright green. Remove broccoli from heat, plate with the steaks and serve.