In a small saucepan, combine water, vinegar, sugar and 1/2 teaspoon salt. Bring to a boil; whisk until sugar is dissolved. Remove from heat. Add carrots and radish; let stand until serving. Shape beef into four 3/4-in.-thick patties. Sprinkle with pepper and remaining 1/2 teaspoon salt.
In a 12-in. cast-iron or heavy skillet, cook burgers over medium heat until a thermometer reads 160°, 8-10 minutes on each side. Remove and keep warm. Wipe pan clean. In the same pan, toast rolls over medium heat, cut side down, 30-60 seconds. Spread rolls with mayonnaise. Drain pickled vegetables. Serve burgers on rolls with picked vegetables, cucumber and cilantro.
