Whisk 2 tbsp. soy sauce, 1 tbsp. oil, 1 tsp. ginger and the white and pale-green parts of the scallions in a large bowl. Add the steak to the marinade and turn to coat. Bring a kettle of water to a boil.
Bring the broth, the remaining 2 tbsp. soy sauce and 2 tsp. ginger to a boil in a 6-qt. pot. Reduce the heat to low and simmer while you cook the steak.
Heat the remaining 1 1/2 tsp. oil in a large nonstick skillet over medium-high. Remove the steak from the marinade, scraping the scallion pieces back into the bowl. Add the steak to the skillet; cook to desired doneness, about 4 minutes per side for medium-rare. Transfer the steak to a plate. Pour the scallion marinade into the skillet; stir to heat through and add to the soup pot.
Add the noodles in a large bowl and cover with boiling water. Cover the bowl with a plate and let the noodles sit while you cook the vegetables.
Add the broccoli to the soup pot; cook 3 minutes. Add the pepper and corn, and simmer until crisp-tender, about 2 minutes. Slice the steak against the grain.
Divide the noodles among bowls using tongs. Top with the vegetables, broth and steak. Sprinkle with the dark-green scallions.