Place the steak and soy sauce in a large resealable plastic bag. Refrigerate for 2 to 8 hours.
Heat grill to medium or place a grill pan over medium heat. Cook the steak to the desired doneness, about 5 minutes per side for medium-rare. Transfer to a cutting board to rest for 10 minutes.
Meanwhile, prepare the noodles according to the package directions. Rinse the noodles under cold water and drain.
In a small bowl, whisk together the ginger (if using), lime juice, brown sugar, salt, and sesame and peanut oils.
Thinly slice the steak across the grain. Divide the steak and noodles among individual plates. Top with the radishes and papaya, drizzle with the dressing, and sprinkle with the cilantro (if using).
Tip: Look for a papaya that has a smooth, dark green skin with a touch of yellow. It should give slightly when squeezed. If you can find only firm papaya, place the fruit in a paper bag and leave it on a counter for a couple of days.