Preheat oven to 400 degrees F.
Take the beef tenderloin and drizzle with olive oil. Season well all over with plenty of kosher salt and freshly ground black pepper all over. Make sure it is at room temperature before cooking. Place a large cast iron skillet over medium-high heat on the stove top and sear tenderloin all over. Place the whole pan in the oven and roast the tenderloin for 15-17 minutes (depending on how thick your tenderloin is). An instant-read thermometer when inserted into the thickest part of the meat should read 130 degrees F for medium-rare. If you want it medium, cook for a further 3 minutes in the oven. Remove from the oven and tent with foil while you let it rest for 10 minutes before cutting.
Make the bearnaise reduction first. In a small saucepan, combine half of the tarragon, shallots, vinegar and wine over medium-high heat. Bring to a simmer and cook until reduced by half. Remove from heat and set aside to cool.
Blend yolks and bearnaise reduction together in a blender. With the blender running, add 1/3 of the butter in a slow steady stream. Once it emulsifies, turn the blender speed up to high and add the remaining butter. Add the remaining half bunch of fresh tarragon leaves, season with salt and pepper and give it 1 more buzz. Set aside in a warm spot to hold the sauce.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.