Although this dinner is free of gluten, dairy, soy, and nuts, no one who eats it will feel deprived.
Heat broiler with rack in second highest position. In a medium pot, bring potatoes to a boil in salted water over high. Reduce heat and cook at a rapid simmer until tender when pierced with a knife, 8 minutes. Drain potatoes; halve lengthwise. Place steak on a broiler-proof rimmed baking sheet. Season with salt and pepper. In a small bowl, combine 1 tablespoon plus 2 teaspoons oil, rosemary, and garlic, then pour over steak, turning to coat. Broil until steak is medium-rare, 7 minutes, flipping once. Transfer to a cutting board, tent with foil, and let rest 10 minutes before slicing against the grain.
Meanwhile, in pot, heat 1 1/2 teaspoons oil over medium-high. Add potatoes, season with salt and pepper, and cook, stirring occasionally, until browned and crisp, 8 minutes. In a medium bowl, whisk together vinegar and 2 teaspoons oil; season with salt and pepper. Add potatoes and arugula; toss to combine. Serve with steak.