Place steak in a medium bowl and sprinkle with flour; turn to coat. Heat oil in a large saucepan over medium-high heat. Add the steak (reserving excess flour) and cook, stirring once or twice, until browned on most sides and still pink in the center, about 3 minutes. Transfer to a plate and tent with foil to keep warm.

Add mushrooms, onions and tomatoes to the pan and cook, scraping up any browned bits, until the vegetables have released their juices, about 3 minutes. Sprinkle the reserved flour over the vegetables; stir to coat. Add green beans, broth, wine, thyme, salt and pepper; increase the heat to high and bring to a boil, stirring often. Reduce heat to a simmer and cook, stirring often, until the broth has thickened, about 5 minutes. Add the steak and any accumulated juices and cook, stirring often, until heated through, about 2 minutes.