Start by infusing your vinegar for the Béarnaise sauce. Heat the vinegar in a small saucepan over medium-high heat. Add the shallot and 1 tablespoon of the tarragon. Simmer for five minutes, then remove from the heat. Strain, then set aside.
Prep and peel your potatoes. Give them a good rinse under cold water. Heat your water to a low simmer (small bubbles) and then add the vinegar. Add the potatoes, and simmer gently for 10 minutes. This draws the starch out of the potatoes.
Remove the fries to a cooling rack over a baking sheet. Let them cool off and dry off. In the meantime, add your oil to a heavy-bottomed pot. Add just enough oil to cover the fries, ensuring there's lots of room between the oil and the edge of the pot. When you add the fries, the moisture in them will bubble up significantly. You don't want this boil over the edge.
Once your oil is at 400℉, add your fries. Cook for about 1 minute, then remove. They should still be white in color. Place them on a cooling rack over a baking sheet. Let them cool off for 15-20 minutes.
Fry again until the fries are golden and crispy - another 3-4 minutes. Remove again to a bowl lined with paper towels. Season generously with salt, and then place in a 200℉ oven to keep warm while you make the steak, and finish the Béarnaise sauce.
Generously season steaks with salt and pepper.
Heat a cast-iron skillet on high. Add olive oil, then sear steaks 3-4 minutes per side.
Reduce heat to medium. Add butter, garlic, and thyme. Baste steaks for 2-3 minutes.
Let steaks rest for 5 minutes.
As the steak rests, finish your Béarnaise sauce. Add the infused, strained vinegar to the blender with the egg yolks. Heat the butter until it's hot, but not boiling (you can do this in the microwave). Slowly add the butter to the eggs and vinegar as you blend, either in a stand blender or with an immersion blender. Taste for seasoning. Add a tablespoon of chopped fresh tarragon and stir.
Plate the steak with fries, and top with Béarnaise sauce. Garnish with parsley or extra tarragon if desired.