Allow the fillets to come to room temperature for about 30 to 45 minutes.
Preheat the oven to 350 degrees.
Season the fillets with steak seasoning on all sides.
Add the butter and the olive oil to a cast iron skillet, and heat it up over medium heat. Stir together as the butter melts.
Once the butter and oil mixture is hot, place the steaks in the pan and sear for three minutes on each side. Don't disturb the steak while it's searing, as you want it to develop a crust, and this won't happen if you move it around in the pan.
Add the herbs to the pan. While the steak is searing, gently braise the meat with butter using a metal spoon.
Once the steak has seared on both sides, transfer the pan to the oven without flipping the steaks again. Cook in the oven for 4 minutes for a medium rare cook. If you like your meat a little more well done, allow it to cook for an extra minute or two. Trust your intuition. Just note that since this is such a tender cut of meat, it will feel soft to the touch.
Pull the steaks out of the oven, and remove them from the pan. Allow them to rest on a separate plate for at least five minutes. This will help to seal in the juices.
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