The term culotte steak is a little vague. It's either the top sirloin cap, or sliced tri-tip. The variation in terminology is, as far as I know, NOT regional.
In any case, there is only one "right" way to pan cook a beef steak. Not that there aren't variations, and not that the "right" means much either -- which is why I put quote marks around it . I'd say it's "the American informed version of French technique," but "right" has a much better ring to it. Don't you think?
1. Season the steak and let it temper.
2. Pre heat the pan over medium-high heat.
3. Remove the pan from the heat, and immediately add a little oil to it. You may use a light oil, EVOO, or a little oil and a little butter.
4. Swirl the pan and if the oil moves very freely and covers the bottom of the pan, the pan is hot enough. If the oil does not it is not ready. In either case return the pan to the heat -- and when the oil does flow ...
5. Add the meat to the pan. Do not turn the meat, do not slide the meat, do not play with the meat, do not give the meat hard looks.
6. After about 90 seconds shake the pan, and if the meat releases from the pan and moves, it's ready to turn. If it does not shake lose, cook another 30 seconds.
7. Shake the pan again, and if the meat releases, turn it. If the meat does not release, try and loosen it by tapping it on the side with your tongs or spatula. Finally, if you can't make it release gently, take charge and use your tongs or spatula to just turn it, dammit.
8. Repeat the same searing technique for the other side. If the steak is too thick to quick through by searing the sides, or your preference is for medium to well-done meat, reduce the heat under the pan from medium-high to medium after 60 seconds, and cook another 90 seconds, then turn for a last 45 to 60 seconds.
These times are based on the assumption your steaks are about 3/4". And very important: I'm not kidding about the exactness of the times, you really have to stay on top of this kind of cooking. You can't wander away and come back when it smells right. This is "turn off the phone" cooking.
9. When the steak is cooked to medium rare (touch test) remove it. If you had to reduce the heat, you'll have to turn it and cook the first side a little more to make sure the steak is cooked evenly on both sides. Lots of turning isn't a good thing but uneven is worse.
10. That's it. You've got fond in the pan and you're ready to deglaze and make a pan reduction.
By the way: There's a thread running around somewhere with my recipe for steak with a pan deglaze, which has wonderful photographs and commentary by RPMcMurphy who followed and illustrated the recipe. If you can't find it, I'll do it for you or (with luck) he or NRatched will.
Hope this helps,