Season the beef tenderloin generously with salt and pepper. Let it rest while you prepare the fries and béarnaise sauce.
Preheat the oven to 400°F.
Cut the potatoes into thick matchsticks and lightly spray them with olive oil.
Arrange the potatoes on a baking sheet and bake at 400°F until golden and crispy. Season with salt immediately after removing them from the oven.
In a small pot, combine the finely chopped shallot, dry white wine, champagne vinegar, and 2 sprigs of fresh tarragon.
Simmer until the mixture has reduced to about 1½ tablespoons of liquid. Strain and set aside.
Melt half of the butter separately.
Add the room-temperature egg yolks to a blender. With the blender running, slowly stream in the melted butter until emulsified.
Add the remaining room-temperature butter and blend again until silky smooth.
Finish the béarnaise sauce with chopped fresh tarragon and salt.
Heat a skillet over medium-high heat. Sear the beef tenderloin for about 3 minutes per side, around 12 minutes total for medium rare.
Remove the steak from the skillet and let it rest for 10 minutes before slicing.
Serve the sliced beef tenderloin with the crispy fries and generously spoon the warm béarnaise sauce over the steak and fries.