How to Make It
Prepare the Lemon-Herb Butter: place softened butter in the bowl of a stand mixer fitted with a paddle attachment. Beat butter on medium speed until fluffy, about 2 minutes. Add herbs, shallot, lemon zest, lemon juice, anchovy paste, salt, and pepper; beat until all ingredients are blended and smooth.
Wrap Lemon-Herb Butter in a piece of plastic wrap; shape into a log (7 to 8 inches long). Chill log until firm, about 20 minutes.
Prepare the Steak with Brussels Sprouts and Potatoes: Place halved potatoes and water in a microwavable bowl, and cover with plastic wrap. Microwave on HIGH until potatoes are slightly tender, 4 to 5 minutes. Transfer potatoes to a plate, and discard water. Pat potatoes dry with paper towels. Set aside.
Heat a 10- to 12-inch cast-iron skillet over high. Sprinkle steak on both sides with pepper and 1 1/2 teaspoons of the salt, pressing gently into steak to adhere.
Heat olive oil in preheated skillet; add steak to hot oil, and cook undisturbed for 3 minutes. Turn, and cook on other side undisturbed, 3 to 4 minutes, adding butter to skillet, and spooning melted butter over steak while it cooks last 2 minutes, until a thermometer inserted in thickest portion reaches 122°F, for medium-rare. Transfer to a plate, and drizzle with lemon juice; tent with aluminum foil to keep warm. Allow steak to rest 10 minutes before slicing.
Without wiping skillet clean, add potatoes to skillet over medium-high, and cook, stirring occasionally, until potatoes are starting to brown, about 4 minutes. Add Brussels sprouts, onion, and remaining 1/2 teaspoon salt; cook, stirring often, until Brussels sprouts are tender, onions are lightly browned, and potatoes are deeply browned, about 6 minutes.
Slice steak against the grain. Divide potato mixture and steak among plates. Add a pat of Lemon-Herb Butter to each, allowing butter to melt onto steak and vegetables. Serve immediately.