Melt the butter and olive oil together in a large skillet set over medium-high heat until the mixture turns a bit cloudy and bubbly.
Season the beef with salt and pepper to taste.
Place the meat in the pan and do not move it at all for at least 3 minutes. Using tongs, carefully turn the tenderloin on its side and brown it for 3 minutes. Repeat the same browning process on all exposed surfaces of the meat.
Place the tenderloin on a rack in a roasting pan in the oven. Roast the beef 15 minutes for medium-rare, 20 minutes for medium, and 23 minutes for medium-well.
Transfer the chateaubriand to a warmed serving platter, lightly tent it with a single layer of foil, and allow it to rest, untouched, for 15 minutes.
While the tenderloin is resting, make the wine sauce. Sauté the chopped shallots in the leftover pan juices in the skillet until softened and translucent.
Pour the wine into the skillet and bring the sauce to a boil, scraping up all the browned bits on the bottom of the pan.
Continue boiling the sauce until it reduces by half.
Add the demi-glace to the sauce and continue boiling the mixture until it becomes slightly thickened.
Remove the sauce from the heat and stir in the tarragon and softened butter.
Serve the chateaubriand, sliced on the diagonal, with the wine sauce and chateau potatoes or truffle fries.
Enjoy!
Notes
If you do not have demi-glace, though this will not be as good as one, you can reduce a can of top quality beef consomme by half and use as above.
Resting the Chateaubriand is paramount to creating the perfect slice. By cooking with time to rest the meat means juices are released (needed for the sauce) and the fibres in the meat relax thus making the eventual eating extremely tender.
If you have ever wondered whether Chateaubriand is a cut of beef or a recipe, check out which is correct here.