1. Place potatoes in a large saucepan; cover with cold water. Bring water to a boil, and cook 15 minutes until potatoes can be pierced with a fork. Drain and cool potatoes. Slice potatoes in half.
2. While potatoes cook, divide steak into 4 equal portions and season with ï¿½ teaspoon salt and freshly ground black pepper. Heat 2 teaspoons oil in a large heavy skillet over medium-high heat. Add steaks to pan, and cook 4 to 6 minutes per side, or until a meat thermometer inserted into the thickest part of the steak reads at least 145ï¿½. Transfer steaks to a plate and keep warm.
3. Reduce heat to medium-low; add 2 teaspoons oil and shallots. Arrange potatoes, cut sides down, in pan, and cook 5 minutes or until potatoes begin to brown.
4. Season with remaining ï¿½ teaspoon salt and freshly ground black pepper. Transfer potatoes to a bowl and keep warm.
5. Increase heat to medium-high. Add wine and gently scrape pan to loosen browned bits. Add remaining 2 teaspoons olive oil, and cook until liquid is reduced by half. Place 1 cup arugula on each of 4 plates; divide steak and potatoes evenly among them. Top each serving with approximately 1 tablespoon sauce.