Preheat the oven to 275 degrees F. Line a sheet pan with aluminum foil.
Dry the entire tenderloin with paper towels and brush it all over (top and bottom) with the butter. Sprinkle it all over with 4 teaspoons of salt and 2 teaspoons of pepper. It will seem like a lot but trust me, it's important.
Place the filet on the sheet pan and roast it for 1 to 1 1/4 hours, until an instant-read thermometer inserted into the end of the beef registers 130 degrees F for medium rare. Remove from the oven, cover tightly with aluminum foil, and allow to rest at room temperature for 15 minutes. Discard the foil and allow to sit at room temperature.
Meanwhile, place the shallots, 3 tablespoons of tarragon, the wine, and vinegar in a small saucepan and bring it to a boil over medium-high heat. Boil for 5 minutes, until only 1 tablespoon of liquid remains. Set aside to cool for 15 minutes.
Place the egg yolks, lemon juice, mustard, 2 1/2 teaspoons salt, and 1 teaspoon pepper in the bowl of a food processor fitted with the steel blade and process for 10 seconds. With the processor running, slowly pour the vegetable oil and olive oil through the feed tube to make a thick emulsion. Add the shallot reduction and the remaining tablespoon of tarragon leaves and pulse to combine. Refrigerate until ready to serve.
Slice the filet of beef between 1/4 and 1/2 inch thick, sprinkle with salt, and serve warm, at room temperature, or cold with the béarnaise mayonnaise on the side.