Season the flank steak with salt and pepper. In a large pot of boiling, salted water, cook the noodles until al dente; drain.
Meanwhile, in a large skillet, heat the oil over medium-high heat until almost smoking. Add the steak and cook, turning once, for 8 minutes for medium-rare; transfer to a cutting board and tent with foil.
Add the caraway seeds to the skillet and cook over medium-high heat for 1 minute. Add the cabbage and beet and cook, stirring, until wilted, about 5 minutes. Add the vinegar, sugar and 3/4 cup water. Lower the heat to medium-low and cook, stirring occasionally, until the cabbage is tender, about 10 minutes; season with salt and pepper.
Slice the steak thinly against the grain, adding any remaining juices to the cabbage mixture. Divide the noodles among 4 plates and top with the cabbage-beet mixture, steak and dill.