I mentioned this to a steak-loving friend of mine who offered this system which I will try... first step is to buy a cast iron skillet.
I used to use 60 pounds of charcoal a year but when it came to cooking for one I abandoned the grill and took up with high quality cast iron skillets. (Lodge Manufacturing makes the best, in Tennessee, where else?). I got one of their smallest that I use to cook myself a filet a couple of times a week. (Stockyards has the best I have found.) I have bigger skillets for company.
I start with a six-ounce US Choice or Prime completely trimmed filet. The way they cut them, that is about two inches thick. Most of the cooking directions give times based on thickness.
I put a touch of olive oil (prevents sticking) and a dollop of butter (good taste) in the iron skillet and heat it until the butter is melted and starting to bubble just a bit. Then I set the heat at about half and put the steak in (with tongs), the steak having been seasoned with sea salt and a little pepper. Brown the steak for about two minutes and turn it. If it tries to stick when you turn, it needed a little more time. Then two minutes on the other side.
Now the important part: The oven has been preheated to 350. Put the steak (in the skillet) into the oven. For my 6-ounce, 2-inch thick steak I then cook for six minutes, turn, and cook for five more minutes. That leaves it on the rare side of medium rare. Six on each side brings it to medium rare. For thinner steaks you would use a little less time on each side.
The advantage to using the stove top and the oven is that the steak is cooked uniformly all the way through. There is no mushy middle as often happens when you go for medium rare on the stovetop alone. I think you would enjoy rarer steaks that are cooked in this manner.