Place the potatoes in a large pot and add enough cold water to cover. Bring to a boil and add 2 teaspoons salt. Reduce heat and simmer until very tender, 15 to 18 minutes. Drain and return to the pot. Toss in the chives, 2 tablespoons of the oil, 1 tablespoon of the vinegar, and ¼ teaspoon each salt and pepper. Lightly flatten each potato with a spoon. Cover to keep warm.
Meanwhile, combine the tomatoes, sugar, 3 tablespoons water, the remaining 3 tablespoons of vinegar, and ¼ teaspoon each salt and pepper in a medium saucepan over medium-high heat. Cook, stirring occasionally, until the tomatoes are soft and the liquid is thickened, 6 to 8 minutes.
Heat the remaining tablespoon of oil in a large skillet over medium-high heat. Season the steaks with ½ teaspoon salt and ¼ teaspoon pepper. Cook until an instant-read thermometer inserted into the thickest part of each steak registers 130° F, 4 to 5 minutes per side for medium-rare. Let rest for 5 minutes, then slice against the grain.
Serve the tomato vinaigrette over the steak with the potatoes alongside.