Preheat your oven to 375º F.
Liberally sprinkle salt and pepper on the steak.
Preheat an oven-safe heavy skillet or saute pan over medium-high heat. Depending on your stove, this could take 4 or 5 minutes. I set a timer to make sure I'm not rushing.
Add the butter and oil to the pan. When the butter stops foaming, sear the meat on all sides until well browned.
Place the meat in the hot pan, and do not move it for at least 2 minutes. With tongs, turn the meat and continue searing. If the meat sticks to the pan, leave it for another few seconds.
When the sear is complete, the meat will release on its own, so be gentle and patient. Keep an eye on the heat, you may need to adjust it up or down to maintain a good "sizzle ' without burning the meat.
Remove the meat from the pan, and place an oven-safe rack in the cooking pan.
Put the meat on the rack and roast in the oven until the meat has reached an internal temperature of 125º F. Use a probe thermometer so you don't have to keep opening the oven. Alternately, check the internal temperature with an instant read thermometer after ten or twelve minutes. Remove from the oven. Put the meat on a warmed platter to rest for about 15 minutes. The temperature will continue to rise, and your meat will be a perfect medium rare.While the meat is resting, prepare the sauce. You should have plenty of oil/butter left in the cooking pan.
Place the pan over medium heat - careful, it has been in the oven. Make sure you have an oven mitt, because the handle can burn you.
Add the minced shallot and saute until translucent, but not browned.
Add the red wine. Turn the heat up to medium high and reduce by half. Add the demi-glace to the pan and reduce for a couple of minutes until the mixture is somewhat syrupy.
Taste for seasonings, and add salt and pepper if necessary.
Stir in the minced tarragon and remove from the heat. Swirl in the softened butter right before serving. This will help to further thicken the sauce and impart a lovely sheen.For a classic presentation, slice your Chateaubriand in half diagonally and serve on warmed plates with the sauce spooned over.
Garnish with some fresh tarragon leaves.