Put the potatoes in a medium pot and cover with cold water by 1 inch; add a generous pinch of salt. Bring to a boil and cook until tender throughout, about 10 minutes. Drain, cool slightly and then halve or quarter if large.
While the potatoes cook, season the steak generously on both sides with salt and pepper. Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Add the steak and cook, turning once, until brown on both sides and cooked to desired doneness, 4 to 5 minutes per side for medium-rare. Transfer to a cutting board to rest for 5 minutes and reserve the skillet for making the vinaigrette.
Bring 1/4 cup water and the paprika to a boil in the reserved skillet over medium-high heat, scraping up any brown bits, until slightly reduced, about 30 seconds. Remove from the heat and stir in the vinegar, mustard, soy sauce, Worcestershire and the remaining 2 tablespoons oil.
Thinly slice the steak against the grain. Pour any accumulated juices into the skillet with the vinaigrette. Divide the arugula, tomatoes and potatoes among 4 large plates or bowls. Top with the sliced steak and evenly drizzle with the vinaigrette.
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