Heat the oil in a medium, heavy saucepan over medium-high heat until hot. Stir in the mushrooms and cook until golden, about 6 minutes. Stir in the garlic and ginger and cook, stirring until golden, about 5 minutes. Stir in the beef broth, 2 cups water, the soy sauce and 1/2 teaspoon each salt and pepper and bring to a boil.
Cut the steak into thin strips and divide among 4 serving bowls. Place 1/2 cup rice on top. Season the broth with salt and pepper, and then ladle into the bowls. Divide the cucumbers, spring onions, chilli and kimchi among the bowls. Serve with lime wedges.