Set your oven to 375°F. Generously season your beef tenderloin all over with salt and freshly ground black pepper. This step not only enhances flavor but also creates a delicious crust when seared.
Heat 2 tablespoons of butter and olive oil in your skillet over medium-high heat. Wait until it bubbles but doesn’t turn brown. Place the beef in the skillet and sear each side for about 3 minutes, ensuring an even golden crust. This seals in the juices before roasting.
Transfer the seared tenderloin to a rack in your roasting pan. Roast for 15 minutes for medium-rare, or slightly longer depending on your preference. Use a meat thermometer to confirm your desired doneness (medium-rare is 135°F).
Once roasted, transfer the tenderloin to a warm platter and lightly tent it with foil. Resting for 15 minutes redistributes the juices, ensuring the meat is melt-in-your-mouth tender.
While your meat rests, sauté the chopped shallots in the same pan where the beef was seared. Deglaze with red wine, scraping up browned bits for maximum flavor. Reduce this mixture by half over medium heat, then stir in demi-glace and cook until slightly thickened. Finally, whisk in the remaining butter and tarragon to finish the sauce.
Slice the rested tenderloin diagonally into thick medallions. Pour the warm sauce over the slices or serve it alongside as a dip.