Season both sides of the 6oz Filet Mignon with salt and pepper and sear with the oil in a hot pan. In a covered casserole or Dutch oven, spread the onion, carrot and celery in an even layer over the bottom. Place the seared steaks, garlic and thyme on the bed of vegetables and pour the butter over the top. Cover and bake in a 300 degree oven for 30 minutes or until desired doneness. When finished, remove the 6oz Filet Mignon from the dish and discard the butter and vegetables.
Poele is a nearly forgotten French culinary technique called butter braising. It is no longer used primarily because of its expense and the need for an already tender cut of meat.