Preheat your oven to 375, then blot the chateaubriand dry with paper towels, keeping the surface dry will help create a good crust.
Melt 2 tablespoons of butter with olive oil in a large cast iron skillet on medium-high heat until cloudy and bubbly.
Add chateaubriand and brown for 3 minutes without moving. Using tongs, turn and brown on all exposed sides.
Transfer the chateaubriand to a rack placed in a roasting pan and put in the oven. (Set aside the skillet with juices for the sauce.) Roast the beef for 15 minutes for medium-rare, 20 minutes for medium, and 23 minutes for medium-well.
Remove the meat from the oven and transfer to a warm serving platter. Lightly tent the meat with foil and let it rest for 15 minutes.
While the chateaubriand is resting, make the wine sauce.
Pour the wine into the skillet and bring the sauce to a boil, scraping up any browned bits on the bottom of the pan, then continue boiling the sauce until is reduced by half.
Add demi-glace to the sauce and boil the mixture until slightly thickened, then remove from the heat and stir in the remaining butter and tarragon. Season with salt and pepper as needed.
Slice the meat on the diagonal and serve with the wine sauce. Enjoy!