Step 1

Put the peeled potatoes in a bowl of cold water. Slice them lengthwise 1/4 inch thick, then cut lengthwise into 1/4-inch-thick fries. Rinse in cold water until the water runs clear. Refrigerate overnight in cold water.

Step 2

In a large pot, bring the gallon of water to a boil. Stir in the salt and vinegar. Drain the potatoes and add them to the boiling water. Return to a gentle boil and cook for 3 minutes. Drain the potatoes well, then spread them in a single layer on paper towels. Pat thoroughly dry and let cool.

Step 3

Line 2 baking sheets with paper towels. In a large enameled cast-iron casserole, heat 2 inches of oil to 365°. Working in batches, fry the potatoes for 1 minute, stirring gently so they don’t stick together. Using a slotted spoon, transfer the fries to the paper towels to cool.

Step 4

In a medium saucepan, combine the shallots, wine and vinegar and cook over moderate heat until all of the liquid has evaporated, about 5 minutes. Scrape the shallots into a medium bowl. Add the butter and beat with a handheld mixer until creamy. Fold in the herbs and season with salt and pepper.

Step 5

Season the steaks with salt and pepper. In a large cast-iron skillet, heat the oil. Cook 2 of the steaks over moderately high heat, turning once, until browned and medium-rare, 3 to 4 minutes per side. Transfer to plates. Repeat with the remaining 2 steaks. Let rest for 5 minutes.

Step 6

Meanwhile, heat the oil in the casserole to 400°. In 2 batches, fry the potatoes, stirring occasionally, until golden and crisp, about 2 minutes. Drain briefly on paper towels. Season with salt. Top the steaks with some of the béarnaise butter and serve immediately with the fries and the remaining béarnaise butter.