In a large pot, heat 1 tablespoon of the oil over moderately high heat. Add half the steak and cook, stirring frequently, until well browned, about 2 minutes. Remove the steak with a slotted spoon. Repeat with another tablespoon of oil and the remaining steak. Remove the steak from the pot and toss all of the steak slices with 1/4 teaspoon of the salt.

Reduce the heat to moderately low and add the remaining tablespoon of oil to the pot. Add the onion and cook, stirring occasionally, until translucent, about 5 minutes.

Add the potatoes, green beans, water, broth, the remaining 1 teaspoon salt, and the pepper to the pot. Bring to a boil, scraping the bottom of the pot with a spoon to dislodge any brown bits. Reduce the heat and simmer until the potatoes are tender, about 15 minutes. Return the steak and any juices to the soup and stir in the Worcestershire sauce.